Computation of matrices and submodular functions values

finite topological spaces
Julian L. Cuevas Rozo1,∗ , Humberto Sarria Zapata1
1 Cienc. Ex. Fis. Nat. 41(158):127-136, enero-marzo de 2017
Rev. Acad. Colomb.
MatricesBogotá,
associatedColombia
to finite topological spaces
Dpto. de Matemáticas, Facultad de Ciencias, Universidad Nacional de Colombia,
doi: http://dx.doi.org/10.18257/raccefyn.409
Abstract
Original article
Mathematics
The submodular
functions have
shown
their importance functions
in the study and
characterization
of
Computation
of matrices
and
submodular
values
associated
to
multiple properties of finite finite
topological
spaces,
from
numeric
values
provided
by
such
functions
topological
spaces
Computation
of matrices
and
functions
(Sarria, Roa & Varela,
2014). However,
the submodular
calculation of these values
has been values
performed
Computation
of matrices
and submodular
functions values
associated
to
manually or even using
Hasse
diagrams,
which
we
present
some
1,∗ is not practical. In this article,
1
Julian L. finite
Cuevas
Rozo
, Humberto
Sarria spaces
Zapata
associated
to
finite
topological
topological
spaces
1
algorithms
that let us calculate some kind of polymatroid functions, specifically fU , fD , f and rA ,
Dpto. de Matemáticas, Facultad de Ciencias, Universidad Nacional de Colombia, Bogotá, Colombia
which define a topology, by using topogenous1,∗matrices.
Julian L. Cuevas Rozo , Humberto Sarria Zapata1
1
Dpto. de Matemáticas, Facultad de Ciencias, Universidad Nacional de Colombia, Bogotá, Colombia
Key
words: Finite topological spaces, submodular functions, topogenous matrix, Stong matrix.
Abstract
nd submodular functions
values
associated
to
Cálculo
de matrices
y funciones
submodulares
asociadas
espacios
finitos
The
submodular
functions
have shown
their importance
in thea study
and topológicos
characterization
of
Abstract
nd
submodularspaces
functions
values
associated
to from numeric values provided by such functions
te topological
multiple properties
of finite
topological spaces,
te topological
spaces
Resumen
(Sarria,
Roa &1functions
Varela, 2014).
However,
calculationinofthe
these
values
been performed
The submodular
have shown
theirthe
importance
study
andhas
characterization
of
1,∗
vas Rozo , Humbertomanually
Sarria Zapata
or even using
Hasse
diagrams, spaces,
which is
not numeric
practical.values
In this
article,by
wesuch
present
some
multiple
properties
of finite
topological
from
provided
functions
1 Bogotá,
e Ciencias,
Universidad
Nacional
de Colombia,
Colombia
vas
Rozo1,∗
, Humberto
Sarria
Zapata
Las
funciones
submodulares
han
mostrado
su importancia
en el specifically
estudio y flaU , caracterización
algorithms
that&
letVarela,
us calculate
some
kind of polymatroid
functions,
fD , performed
f and rA ,
(Sarria,
Roa
2014).
However,
the
calculation
of
these
values
has
been
e Ciencias, Universidad Nacional
de
Colombia,
Bogotá,
múltiples
propiedades
de
los
espacios
topológicos
finitos, a partir de valores numéricos proporwhich
define
a topology,
byColombia
using
topogenous
matrices.
manually
or even
using Hasse
diagrams,
which
is not practical. In this article, we present some
cionados por dichas funciones (Sarria, Roa & Varela, 2014). Sin embargo, el cálculo de éstos
algorithms that let us calculate some kind of polymatroid functions, specifically fU , fD , f and rA ,
valores se ha Finite
realizado
manualmente
e submodular
incluso haciendo
uso detopogenous
diagramasmatrix,
de Hasse,
lo que
no es
Key
topological
functions,
Stong
matrix.
whichwords:
define a topology,
by usingspaces,
topogenous
matrices.
práctico. En este artı́culo, presentamos algunos algoritmos que nos permiten calcular cierta clase
hown their importance in the study and characterization of
de funciones
polimatroides,
especı́ficamente
fU , fD , fasociadas
y rA , las a
cuales
definen
una topologı́a,
por
Cálculo
de
matrices
y funciones
submodulares
espacios
topológicos
finitos
Key
words:
Finitecharacterization
topological
spaces,
gical spaces,
from numeric
values
provided
by
such
functions
hown
their importance
in
the
study
of submodular functions, topogenous matrix, Stong matrix.
medio
del usoand
de matrices topogéneas.
However,
the from
calculation
these provided
values hasbybeen
gical
spaces,
numericofvalues
suchperformed
functions
Resumen
Cálculo
de
matrices
y
funciones
rams,
which
is
not
practical.
In
this
article,
we
present
somesubmodulares asociadas a espacios topológicos finitos
However, the calculationPalabras
of these values
has
been
performed
clave: Espacios topológicos finitos, funciones submodulares, matriz topogénea, matriz
e
kind
of
polymatroid
functions,
specifically
f
,
f
,
f
and
rA ,
rams, which is not practical.
In this article, Uwe present
some
D
de Stong.
Las
funciones submodulares han mostrado su importancia en el estudio y la caracterización
Resumen
opogenous
matrices. functions,
me
kind of polymatroid
specifically fU , fD , f and rA ,
múltiples propiedades de los espacios topológicos finitos, a partir de valores numéricos proporopogenous matrices.
cionados
por dichas
funciones
(Sarria,
Roa su
& Varela,
2014).
embargo,
de éstos
Lastopogenous
funciones
submodulares
han
mostrado
importancia
en Sin
el estudio
y el
la cálculo
caracterización
ces, submodular functions,
matrix,
Stong matrix.
valores
se
ha
realizado
manualmente
e
incluso
haciendo
uso
de
diagramas
de
Hasse,
lo
que
no es
múltiples
propiedades
los matrix.
espacios topológicos finitos, a partir de valores numéricos proporaces, submodular functions,
topogenous
matrix, de
Stong
práctico.
En
este
artı́culo,
presentamos
algunos
algoritmos
que
nos
permiten
calcular
cierta
clase
cionados
por dichas
funciones finitos
(Sarria, Roa & Varela, 2014). Sin embargo, el cálculo de éstos
s submodulares asociadas
a espacios
topológicos
de funciones
polimatroides,
especı́ficamente
fU ,haciendo
fD , f y ruso
cuales definen
una topologı́a,
por
A , las
valores
se
ha
realizado
manualmente
e
incluso
de
diagramas
de
Hasse,
lo
que
no
es provide
s submodulares
asociadas a espacios topológicos finitos
Introduction
problem still unsolved. Moreover, such matrices
medio del En
usoeste
de matrices
práctico.
artı́culo, topogéneas.
presentamos algunos algoritmos
que
nos
permiten
calcular
cierta
clase
all the information about the topology of a finite space
de funciones polimatroides, especı́ficamente fU , fD(Shiraki,
, f y rA , las
cuales
definen
topologı́a,
por objects in
1968),
showing
theuna
relevance
ofmatriz
these
Palabras
clave:
Espacios
topológicos
finitos,
funciones
submodulares,
matriz
topogénea,
Alexandroff
proved
that
finite
topological
spaces
are
medio
del
uso
de
matrices
topogéneas.
mostrado su importancia en el estudio y la caracterización
the topological context.
Stong. valores
in
correspondence
one-to-one
with proporfinite preorders
ios topológicos
finitos,
adepartir
mostrado
su importancia
en el de
estudio
y numéricos
la caracterización
(Alexandroff,
1937),
showing
thattopológicos
such
spacesfinitos,
can be
clave:
Espacios
funciones submodulares, matriz topogénea, matriz
ria,
Roa & Varela,
Sin
el cálculo
de
éstos
ios topológicos
finitos,2014).
aPalabras
partir
deembargo,
valores
numéricos
proporRecently, connections between submodular functions and
viewed
from
other
mathematical
structures.
Likewise,
Shide
Stong.
eria,
e incluso
haciendo
uso
de
diagramas
de
Hasse,
lo
que
no
es
Roa & Varela, 2014). Sin embargo, el cálculo de éstos
finite topological spaces have been developed (Abril,
raki named
topogenous
objects
mos
algunoshaciendo
algoritmos
permiten
calcular
cierta
clase
te
e incluso
usoque
deasnos
diagramas
dematrices
Hasse, lothe
que
no
es worked by 2015), (Sarria et al., 2014), allowing to interpret many
Krishnamurthy,
inpermiten
andefinen
attempt
to topologı́a,
count
theclase
topologies
that
camente
fU , falgoritmos
las nos
cuales
una
por
mos algunos
calcular
cierta
D , f y rA ,que
topological concepts through numeric values provided by
can
defined
on a finite
setuna
(Krishnamurthy,
as.
camente
fU , fDIntroduction
, f yberA
, las cuales
definen
topologı́a, por 1966), a problem
such associated
functions,
which issuch
reallymatrices
important
if we
still unsolved.
Moreover,
provide
eas.
want
to
mechanize
the
verification
of
topological
properall
the
information
about
the
topology
of
a
finite
space
* Correspondence: J. L. Cuevas Rozo, [email protected],
cos finitos, funciones
submodulares, matriz topogénea, matriz
Received xxxxx XXXX; Accepted xxxxx XXXX.
ties on subsets
ofshowing
an arbitrary
finite
space.
(Shiraki,
1968),
the relevance
ofmatrices
these objects
in
Introduction
still
unsolved.
Moreover,
such
provide
Alexandroff
proved that
finite
topological
icos finitos, funciones
submodulares,
matriz
topogénea,
matrizspaces are problem
the
topological
context.
In
view
of
the
above,
we
regard
some
matrices,
as
toin correspondence one-to-one with finite preorders all the information about the topology of a finite space
pogenous and Stong matrices defined in sections 2 and 3,
(Alexandroff,
1937),that
showing
such spaces
can are
be (Shiraki, 1968), showing the relevance of these objects in
Alexandroff
proved
finitethat
topological
spaces
which areconnections
useful, for between
example,submodular
in the study
of lattice
Recently,
functions
andof
viewed
from other mathematical
Likewise,
Shi- the topological context.
in
correspondence
one-to-one structures.
with finite
preorders
all topologies
on aspaces
fixed have
set, and
we developed
link such matrices
finite
topological
been
(Abril,
raki named as topogenous
matrices
objects
worked
(Alexandroff,
1937), showing
thatthesuch
spaces
can by
be 2015),
rA ),
ffunctions
with particular
submodular
(f
(Sarria
et
al.,between
2014),functions
allowing
toU ,interpret
many
D , f and
Recently,
connections
submodular
and
Krishnamurthy,
in
an
attempt
to
count
the
topologies
viewed from other mathematical structures. Likewise, that
Shi- topological
important
in
the
finite
topological
spaces
context,
by
alconcepts
through
numeric
values
provided
by
finite topological spaces have been developed (Abril,
problem
stillasunsolved.
such
matrices
provide
can be
defined
on a finiteMoreover,
set
(Krishnamurthy,
1966),
raki
named
topogenous
matrices
the objects
worked
bya such
gorithms
created
to
improve
the
computations
shown
in
associated
functions,
which
is
really
important
if
we
2015), (Sarria et al., 2014), allowing to interpret many
all
the information
the to
topology
of topologies
a finiteprovide
space
problem
still unsolved.
Moreover,
suchthe
matrices
Krishnamurthy,
in anabout
attempt
count
that want
(Abril,
2015)
and
(Sarria
et
al.,
2014).
to
mechanize
the
verification
of
topological
proper*
Correspondence:
J.
L.
Cuevas
Rozo,
[email protected],
(Shiraki,
1968),onshowing
objects
ina topological concepts through numeric values provided by
information
aboutthe
therelevance
topologyofofthese
a finite
space
canthe
be defined
a finite
set
(Krishnamurthy,
1966),
spaces are all
Received
xxxxx XXXX;
Accepted
xxxxx
XXXX.
ties on
subsets offunctions,
an arbitrary
finite
space.important if we
which
is really
the
topological
context.
(Shiraki,
1968),
showing
the
relevance
of
these
objects
in such
Fromassociated
now on, we
shall use
exclusively
the symbol X to
e spaces
preorders
are
want
to amechanize
the verification
of1 ,topological
properthe
topological
context.
*
Correspondence:
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L.
Cuevas
Rozo,
[email protected],
.
.
.
, xn }, unless
exdenote
set
of
n
elements
X
=
{x
can be
eaces
preorders
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on subsets
of an arbitrary
finite Ispace.
Recently,
connections
betweenxxxxx
submodular
=
{1,
2,
.
.
.
,
n}
and
plicitly
stated
otherwise.
We
define
n
ikewise,
aces canShibe
*Corresponding autor:
finite topological spaces have been developed (Abril,
a permutation
σ of In , we use Pσ to denote the matrix
Julian
L. Cuevas Rozo, [email protected]
tsikewise,
workedShiby Recently, connections between submodular functions and for
2015),
(Sarria et spaces
al., 2014),
to interpret(Abril,
many whose
finite topological
haveallowing
been developed
entries
satisfy
Received:
September
30, 2016
pologies
that
ts worked
by
topological
concepts
through
numeric
values
provided
by
Accepted: March 6, 2017
2015),
(Sarria
et
al.,
2014),
allowing
to
interpret
many
hy, 1966),
a
pologies
that
such
associated
functions,
which
is really
important
if we
[Pσ ]ij = δ(i, σ(j))
topological
concepts
through
numeric
values
provided
by
hy, 1966), a
to mechanize
the verification
topological
properExampl
such associated
functions,
which is of
really
important
if we
[email protected], want
127 where X
of anthe
arbitrary
finiteofspace.
wantontosubsets
mechanize
verification
topological proper- where δ is the Kronecker delta.
[email protected], ties
ties on subsets of an arbitrary finite space.
T =
, as to2 and 3,
attice of
matrices
and rA ),
t, by alhown in
ol X to
nless ex, n} and
e matrix
1 0 0 0 1 0
which are useful, for example, in the study of lattice of
6
. . , xsuch
exdenote
a set of
= {x
n }, unless
0 1 0 •x
0 0 0
all
topologies
onnaelements
fixed set,Xand
we1 , .link
matrices


2, . . . , n} and
plicitly
stated otherwise.
Wefunctions
define In(f= ,{1,
•x2 
0 0 1 0 0 0
with
particular
submodular
U fD , f and rA ),


T
=
for a permutation
σ of Itopological
the matrix
X
n , we use Pspaces
σ to denote
0 1 0 1 0 1.
important
in the finite
context,
by al Fis. Nat. 41(158):127-136,enero-marzo de 2017
Cuevas
Rozo
JL, Sarria
Zapata H
Rev. Acad. Colomb. Cienc. Ex.
whose
entries
satisfy
1 0 0 0 1•x05
gorithms created to improve the computations shown in
doi: http://dx.doi.org/10.18257/raccefyn.409
•x
0 03 0 0•x 0 1
(Abril, 2015) and (Sarria et al., 2014).
1
[Pσ ]ij = δ(i, σ(j))

 space (X, T )
the topological
From now on, we shall use exclusively the symbol X to Example 0.4. Consider
1 0 0 0 1 0
where δa issetthe
and
denote
of Kronecker
n elementsdelta.
X = {x1 , . . . , xn }, unless ex- where X = {a, b, c, d,e}

0 1 0 0 0 0
plicitly stated otherwise. We define In = {1, 2, . . . , n} and
0 0 1 0 0 0
T = {∅, {b}
, {b, d} , {d, e} , {b, d,e} , {a, b, d},
TX, {d}
=
for a permutation σ of In , we use Pσ to denote the matrix
0 1 0 1 0 1.

 b, c, d} , X}
b, d, e} , {a,
whose
entries satisfy matrix
1 0 {a,
Topogenous
0 0 1 0
0 0 of0 the
0 elements,
0 1
For the following orderings
we obtain the
[Pσ ]ij = δ(i, σ(j))
Given a finite topological space (X, T ), denote by Uk the Example
respective topogenous
matrices
as can be verified
by T
cal0.4. Consider
the topological
space (X,
)
minimal
set containing
xk :
culating
the{a,
minimal
basis
where
δ isopen
the Kronecker
delta.
where
X=
b, c, d, e}
andin each case:
(x1,,x
) =e}(a,
b, c,
2, x
3 , xd}
4, x
T = {∅, {b}
{d}
, {b,
, 5{d,
, {b,
d, d,
e}e)
, {a, b, d},
Uk =
E


xk ∈E∈T
1 {a,
0 b,1d, e}
0 , {a,
0 b, c, d} , X}
Topogenous matrix
 1 1 1 0 0


and consider the colection U = {U1 , . . . , Un }, which is the For the following T

orderings
we obtain the
0 0
X1 = 0 of0the1elements,

minimal
basis topological
for the space
in the
sense
U isby
contained
Given
a finite
space
(X,
T ),that
denote
Uk the respective topogenous matrices
1 0 as

can
be
verified
by cal1 1 1
in any other
for the topology
T.
minimal
openbasis
set containing
xk :
culating the minimal basis0 in 0each
0 case:
0 1
Definition 0.1. Let (X, T )
be a finite topological space
(x1 ,x2 , x3 , x4 , x5 ) = (a, b, c, d, e)
= minimalEbasis. The topogenous
and U = {U1 , . . . , UnU}k its


xk ∈E∈T
1 0 1 0 0
to X is the square matrix of
matrix TX = [tij ] associated
(x1 ,x2 , x3 ,x4 , x5 ) = (b, d, e,a, c)
size consider
n × n that
1 1 1 0 0
and
thesatisfies:
colection U = {U1 , . . . , Un }, which is the
01 00 10 01 01
T
=
X
1


in the sense that U is contained
minimal basis for the space
10 01 11 11 11

1 , xi ∈ Uj
in any other basistijfor
= the topology T .
0 00 01 00 10
TX 2 = 
0


0 , in other case
 0 0 0 1 1
Definition 0.1. Let (X, T ) be a finite topological space
0 0 0 0 1
and U = {U1 , . . . , Un } its minimal basis. The topogenous
Remark
0.2.
(Shiraki,
1968)
introduce
the
term
topogematrix TX = [tij ] associated to X is the square matrix of
(x1 ,x2 , x3 , x4 , x5 ) = (b, d, e, a, c)
nousn ×
matrix
denote the transpose matrix of that in
 be characterized

size
n thattosatisfies:
Topogenous matrices can
1 0 0 1 1 by the following
above definition.
result.
0 1 1 1 1


1 ,diagram,
xi ∈ Uj we represent the minExample 0.3. tIn
the
next

0 1 Let
0 T0
ij =
X2 = 0 1968)
Theorem 0.5. T(Shiraki,
= [tij ] be the to0
,
in
other
case
imal basis for a topology on X = {x1 , x2 , x3 , x4 , x5 , x6 }.
0 0to (X,
0 1T ).1Then T satisfies
pogenous matrix associated
The minimal open sets are U1 = U5 = {x1 , x5 } , U2 =
the following properties, 0for 0all 0i, j,0k ∈1In :
, U3 =
{x3 } , U41968)
= {xintroduce
{x4 ,term
x6 } and
the
{x2 , x4 } 0.2.
4 } , U6 = the
Remark
(Shiraki,
topogeassociated
is as follows:
1. tij ∈ {0, 1}.
nous
matrixtopogenous
to denote matrix
the transpose
matrix of that in
2. tii = 1.matrices can be characterized by the following
Topogenous
2. tii = 1.
above definition.
result.
3. tik = tkj = 1 =⇒ tij = 1.
3. tik = tkj = 1 =⇒ tij = 1.
Example 0.3. In the next diagram, we represent the minbe the
to-n
Theorem 0.5. (Shiraki, 1968) Let T = [tij ] size
n×
imal basis for a topology on X = {x1 , x2 , x3 , x4 , x5 , x6 }. Conversely, if a square matrix T = [tij ] of
Conversely,
if a associated
square matrix
T T=). [tThen
size
n×n
pogenous
matrix
to
(X,
satisfies
ij ] of T
The minimal open sets are U1 = U5 = {x1 , x5 } , U2 = satisfies the above properties, T induces a topology on X.
satisfies
the above
properties,
the
following
properties,
for allTi,induces
j, k ∈ Ina: topology on X.
{x2 , x4 } , U3 = {x3 } , U4 = {x4 } , U6 = {x4 , x6 } and the
Homeomorphism
associated topogenous matrix is as follows:
1. tij ∈ {0, 1}. classes are also described by similarity
Homeomorphism
classes are also described by similarity
via a permutation matrix between topogenous matrices.
via a permutation matrix between topogenous matrices.
Theorem 0.6. (Shiraki, 1968) Let (X, T ) and (Y, H )
•x4
Theorem 0.6. (Shiraki, 1968) Let (X, T ) and (Y, H )
be finite topological spaces with associated topogenous mabe finite topological spaces with associated topogenous ma•x6
trices TX and TY , respectively. Then (X, T ) and (Y, H )
trices TX and TY , respectively. Then (X, T ) and (Y, H )
are homeomorphic
spaces if, and only if, TX and TY are
are homeomorphic spaces if, and only if, TX and TY are
•x2
similar via a permutation matrix.
similar via a permutation matrix.
•x5
•x3
Transiting Top(X)
•x1
Transiting Top(X)


1 0 0 0 1 0
Using topogenous matrices, we can find a way to transit
 0 1 0 0 0 0
Using topogenous matrices, we can find a way to transit


through Top(X), the complete lattice of all topologies on
 0 0 1 0 0 0
through Top(X), the complete lattice of all topologies on
.
a fixed set X. Given two topologies T1 and T2 in Top(X),
TX = 
 0 1 0 1 0 1
a fixed set X. Given two topologies T1 and T2 in Top(X),


the supremum of them, denoted by T1 ∪T2 , is the topol 1 0 0 0 1 0
the supremum of them, denoted by T1 ∪T2 , is the topology whose open sets are unions of finite
intersections of
ogy
whose open sets are unions of finite intersections of
0 0 0 0 0 1
elements in the collection T1 ∪ T2 .
elements in the collection T1 ∪ T2 .
Example 0.4. Consider the topological space (X, T )
Proposition 0.7. (Cuevas, 2016) Let (X, T1 ) and
Proposition 0.7. (Cuevas, 2016) Let (X, T1 ) and
where X = {a, b, c, d, e} and
(X, T2 ) be finite topological spaces with minimal basis
128
(X, T ) be finite topological spaces with minimal basis
U = 2{U1 , . . . , Un } and V = {V1 , . . . , Vn }, respectively.
T = {∅, {b} , {d} , {b, d} , {d, e} , {b, d, e} , {a, b, d},
U = {U1 , . . . , Un } and V = {V1 , . . . , Vn }, respectively.
Then, the minimal basis W = {W1 , . . . , Wn } for the space
Then, ∗the minimal ∗basis W = {W1 , . . . , Wn } for the space
{a, b, d, e} , {a, b, c, d} , X}
(X, T ), where T = T ∪ T , satisfies W = U ∩ V
which conta
which
conta
[TX ∗ ]ik = m
[TX ∗ ]ik = m
The second
The second
the minimal
the minimal
by the above
by the above
T(X,T1 ∪T2 )
T(X,T1 ∪T2 )
A way to go
A way to go
is possible u
is possible u
next result.
next result.
Corollary
Corollary
finite topolo
finite topolo
[TX2 ]ij [TX
[TX2 ]ij [TX
Proof. The
Proof. The
lences:
lences:
T1 ⊆ T2 ⇐
T1 ⊆ T2 ⇐
⇐
⇐
Triangula
Triangula
Let (X, T ) b
size n × n
size on
n ×X.
n
ology
ology on X.
by similarity
by
similarity
s matrices.
s matrices.
and (Y, H )
and (Y, maH)
ogenous
ogenous
and (Y, maH)
and (Y,
TY H
are)
and TY are
∗
∗
Using
find a way
to Utransit
(X, T topogenous
), where Tmatrices,
= T1 ∪we
T2can
, satisfies
Wk =
k ∩ Vk
through
Top(X),
the
complete
lattice
of
all
topologies
on
for all k ∈ In .
a fixed set X. Given two topologies T1 and T2 in Top(X),
theTop(X),
topolthe
of them,
denoted
by
T1 ∪T
2 , is in
Thesupremum
next
theorem
way
to move
ahead
Rev.
Acad.
Colomb.
Cienc.shows
Ex. Fis. a
Nat.
41(158):127-136,
enero-marzo
de 2017
ogy
whose
open
sets
are
unions
of
finite
intersections
of
that
is,
it
allows
to
find
the
supremum
of
two
topologies.
doi: http://dx.doi.org/10.18257/raccefyn.409
T2following
.
elements
in the
1∪
The symbol
∧ iscollection
regardedTin
the
sense: if E and
F are n × n matrices,
E ∧ F is 2016)
the square
matrix
Proposition
0.7. (Cuevas,
Let (X,
T1 ) whose
and
entries
satisfy
[E
∧
F
]
=
min
{[E]
,
[F
]
}.
ij
spacesij withij minimal basis
(X, T2 ) be finite topological
Let (X, T ) be a finite topological space with minimal basis
lences:
U = {U1 , . . . , Un } and associated topogenous matrix TX =
]. ⊆Define
theTbinary
relation on X as follows:
[tT
ij 1
T2 ⇐⇒
2 = T1 ∪ T2 The second part of the theorem holds by uniqueness of
The minimal
second part
theorem holds
by(1)
uniqueness
of
the
basisofforthe
a topology,
since if
is satisfied,
thethe
minimal
for awe
topology,
since
by
above basis
argument
would have
TXif∗ (1)
= Tis
∧ TX2 =
X1satisfied,
by(X,T
the1above
argument
would
have
T
thus Twe∗ =
T1 ∪
T2 .TX ∗ = TX1 ∧ TX2 =
∪T2 ) and
T(X,T1 ∪T2 ) and thus T ∗ = T1 ∪ T2 .
•x2
•x4
•x2
•x4
•x2
•x4
In the example 0.4, it was possible to associate an upper
triangular
topogenous
to the considered
since
In the example
0.4, it matrix
was possible
to associatespace
an upper
such
space
is T0possible
, to
as the
shows
the next
Shiraki’s
triangular
topogenous
considered
space
since
In thetopological
example
0.4,
it matrix
was
to associate
an
upper
theorem.
such
topological
spacematrix
is T0 , to
as the
shows
the next
Shiraki’s
triangular
topogenous
considered
space
since
theorem.
such
topological space is T0 , as shows the next Shiraki’s
Theorem
theorem. 0.12. (Shiraki, 1968) A finite topological
if, and only
if, its
topogespace (X, T 0.12.
) is T0(Shiraki,
Theorem
1968)
A associated
finite topological
is
similar
via
a
permutation
matrix
to an
nous
matrix
T
if,
and
only
if,
its
associated
topogespace
(X,
T
)
is
T
X
Theorem 0.12. 0(Shiraki, 1968) A finite topological
upper
triangular
topogenous
matrix.
similar
via
a
permutation
matrix
to an
nous
if,
and
only
if,
its
associated
topogespace matrix
(X,
T )TX
is is
T
0
upper
triangular
topogenous
similar viamatrix.
a permutation matrix to an
nous matrix
TX is
upper triangular topogenous matrix.
A procedure to triangularize a topogenous matrix of a T0
space
X is described
below. aGiven
a topogenous
matrix
A procedure
to triangularize
topogenous
matrix of
a T0
n
space
X
is
described
below.
Given
a
topogenous
matrix
A
procedure
to
triangularize
a
topogenous
matrix
of
a
TX = [tij ] define Mk = n tik = |Uk |, for each k ∈ In ;Tif0
Given a topogenous matrix
i=1
space
described
below.
=X
[tijis
] define
=
|Uk |, for each k ∈ In ; if
TX organize
n tik =
we
them M
inkascending
order
TX = [tij ] define Mk = i=1 tik = |Uk |, for each k ∈ In ; if
we organize them in ascending
order
i=1
we organize themMin
ascending
order
(3)
k1 Mk2 · · · Mkn
Mk1 Mk2 · · · Mkn
(3)
Mk1 Mk2 · · · Mk1n 2 · · · n(3)
,
and consider the permutation σ = k11 k22 · · · knn
,
and consider the permutation
σ
=
the topogenous matrix PσT TX Pσ is upper
k11 k2triangular:
· · · knn if
2
and consider the permutation
σ
=
,
T would have
= 1, P
we
i > jtopogenous
and tσ(i)σ(j)
k1xσ(i)
ktriangular:
· σ(j)
· kand
the
matrix
2 < ·x
n if
σ TX Pσ is upper
T would
M=kj1,which
the
of
we
have
xσ(i)
<ordering
xσ(j) and
itherefore
> jtopogenous
andM
tσ(i)σ(j)
ki <matrix
the
P
Pσ is
upper
triangular:
if
Xcontradicts
σ T
therefore
contradicts
the <ordering
of
k , jhence
iM
>
andM
ttσ(i)σ(j)
=kj0.
1,which
we would
have xσ(i)
xσ(j) and
σ(i)σ(j)
ki < M
tσ(i)σ(j)
=kj0.which contradicts the ordering of
M
therefore
k , henceM
ki < M
Example
Consider
tσ(i)σ(j)
= 0. the topological space X, repreMk , hence 0.13.
129
sented
by
the
next
Hasse
and topogenous
Example 0.13. Considerdiagram
the topological
space X,matrix:
represented by the
nextConsider
Hasse diagram
and topogenous
Example
0.13.
the topological
space X,matrix:
represented by the next Hasse diagram and topogenous matrix:
This relation is a preorder on the space, that is, it is reflexive and transitive. The next theorem shows under which
Triangularization
of topogenous matrices
. , Un }Letand
V
=
{V
,
.
.
.
,
V
},
respectively.
UTheorem
= {U1 , . .0.8.
1
n
X1 = (X, T1 ), X2 = (X, T2 ) and condition such a relation is a partial order on X.
. . , Wnwith
} fortopogenous
the space
Then,
basis topological
W = {W1 , .spaces
(X,minimal
T ∗ ) be finite
X ∗ = the
Theorem
0.10.
(Alexandroff,
1937)with
Topologies
a fiLet
(X, T ) be
a finite
topological space
minimalonbasis
∗
(X,
T ∗ ), where
= TT1X∪∗ ,Trespectively.
∩V
2 , satisfies W
matrices
TX1 , TTX2 and
If kT=∗ U
=k T
1 k∪ Unite
set
X
are
in
one-to-one
correspondence
with
preorders
= {U1 , . . . , Un } and associated topogenous matrix TX =
for
k ∈ In .
T2 allthen
Moreover,
a finite
topological
T ) is T0 if,
Define
the binary
relation
on space
X as (X,
follows:
[ton
ij ].X.
(1) and only if, (X, ) is a poset.
TX ∗ = TX1 ∧ TX2
The next theorem shows a way to move ahead in Top(X),∗
xi xj ⇐⇒ xi ∈ Uj ⇐⇒ Ui ⊆ Uj ⇐⇒ tij = 1 (2)
Conversely,
if there
exist
finite
spacesofXtwo
1, X
2 and X
that
is, it allows
to find
the
supremum
topologies.
Example 0.11. Consider the space (X, T ) given in the
∗
which
satisfy∧ (1)
then T in=the
T1following
∪ T2 . sense: if E and example 0.4 with the ordering
The
symbol
is regarded
F are n × n matrices, E ∗∧ F is the square matrix whose This relation is a preorder on the space, that is, it is reflex= T
Proof. satisfy
Suppose
1 ∪ T
2 and fix an index
(x1 , x2 , x3 , x4 , x5 ) = (a, b, c, d, e).
entries
[E that
∧ F ]ijT= min
{[E]
ij , [F ]ij }.
k ∈ In . By proposition 0.7, we know that Wk = Uk ∩ Vk , ive and transitive. The next theorem shows under which
is aaxiom.
partial order on X.
Theorem
0.8.
Let X = (X, T ), X = (X, T ) and condition such a relation
T
thus xi ∈ W
k ⇐⇒ xi ∈1 Uk and xi1∈ Vk .2 Since the 2column Such space satisfies
0 , x , x } Hasse diagram of the
U1the
= {x
1
2
4
∗
∗
T ) be finite
topological
spaces
with topogenous
Xk of=a(X,
topogenous
matrix
represents
the minimal
open set Theorem
poset (X, )
is the
next:
0.10.
(Alexandroff,
1937)
Topologies on a fiU
,
x
,
x
∗
1
1
2
4}
=
{x
}
2
2
∗
matrices TX1 , TX2 and TX , respectively. If T = T1 ∪ nite set X are in one-to-one correspondence
with preorders
U12 =
= {x
{x12 },, x
2 , x4 }x }
T2 then
3
1 x2 , x3 , space
4
on X. Moreover, a U
finite
topological
(X, T ) is T0 if,
U
{x
(1) and only if, (X, ) U
TX ∗ = TX1 ∧ TX2
=poset.
{x124 }}, x2 , x3 , x4 }
is324 a=
U34 =
{x14 ,,}x
2 ,}x3 , x4 }
U
Conversely, if there exist finite spaces X1 , X2 and X ∗ Example 0.11. Consider
5 = {xthe
4 x5space
(X, T ) given in the
∗
U
= {x
{x44 ,}x5 }
which satisfy (1) then T = T1 ∪ T2 .
U45ordering
=
example 0.4 with the
U5 =
, x5 }
•x{x
3 4
Proof. Suppose that T ∗ = T1 ∪ T2 and fix an index
(x1 , x2 , x3 ,•x
x4 , x5 ) = (a, b, c, d, e).
3
k ∈ In . By proposition 0.7, we know that Wk = Uk ∩ Vk ,
•x3
thus xi ∈ Wk ⇐⇒ xi ∈ Uk and xi ∈ Vk . Since the column Such space satisfies the T0 axiom. Hasse diagram of the
•x5
•x1
k of a topogenous
represents
openthat
set poset (X, ) is the next:
then for
each i ∈ the
In itminimal
is satisfied
which
contains xk ,matrix
which
contains
x
,
then
for
each
i
∈
I
it
is
satisfied
that
•x5
•x1
[TX ∗ ]ik = min {[TkX1 ]ik , [TX2 ]ik }.
n
[TX ∗ ]ik = min {[TX1 ]ik , [TX2 ]ik }.
•x5
•x1
A way to go back in Top(X) is finding subtopologies, which
A way
to gousing
back in
Top(X) is matrices
finding subtopologies,
which
is
possible
topogenous
as is shown in
the
is possible
next
result.using topogenous matrices as is shown in the
next result.
Corollary 0.9. Let X1 = (X, T1 ) and X2 = (X, T2 ) be
(X, T
and
(X,only
T2 ) if,
be
Corollary
0.9. Let
X1 =Then
finite
topological
spaces.
T11) ⊆
T2Xif,
2 =and
finite
topological
spaces.
Then
T
⊆
T
if,
and
only
if,
]
[T
]
for
all
i,
j
∈
I
.
[T
1
2
X2 ij
X1 ij
n
[TX2 ]ij [TX1 ]ij for all i, j ∈ In .
Proof. The result follows from the next chain of equivaay to transit lences:
Proof. The result follows from the next chain of equivaay
to transit
opologies
on lences:
on
in Top(X),
T1 ⊆ T2 ⇐⇒ T2 = T1 ∪ T2 2opologies
Top(X),
TT2 ⇐⇒ [T ] [T ]
T1 ⊆ T2 ⇐⇒
topol2isinthe
1 ∪∧
⇐⇒ T
TX2 2==T
TX
X2
X2 ij
X1 ij
1
is the topolersections
of
⇐⇒ TX2 = TX1 ∧ TX2 ⇐⇒ [TX2 ]ij [TX1 ]ij
ersections of
X, T1 ) and
X, T1 ) basis
and
inimal
inimal
basis Triangularization of topogenous matrices
respectively.
Triangularization of topogenous matrices
respectively.
or the space
Let (X, T ) be a finite topological space with minimal basis
Uk space
∩ Vk
kor=the
Let=(X,
space with minimal
, Uanfinite
} andtopological
associated topogenous
matrix Tbasis
U
{U1T, .). .be
X =
k = U k ∩ Vk
, Un }binary
and associated
matrix TX =
Uij=]. {U
Define
relation topogenous
on X as follows:
[t
1 , . . . the
[tij ]. Define the binary relation on X as follows:
in Top(X),
xi xj ⇐⇒ xi ∈ Uj ⇐⇒ Ui ⊆ Uj ⇐⇒ tij = 1 (2)
Top(X),
o in
topologies.
xi xj ⇐⇒ xi ∈ Uj ⇐⇒ Ui ⊆ Uj ⇐⇒ tij = 1 (2)
ose:
topologies.
if E and
se: if E
and This relation is a preorder on the space, that is, it is reflexmatrix
whose
matrix whose ive
Thisand
relation
is a preorder
on the
space,shows
that is,
it is reflextransitive.
The next
theorem
under
which
ive
and
transitive.
The
next
theorem
shows
under
which
condition
such
a
relation
is
a
partial
order
on
X.
(X, T ) and
2
TX2x=∈TX
∧ TXassociated
[Tfinite
[TX ]ij (2)
X2 ]ijtopological
Matrices
1 ⇐⇒
2
xi ⇐⇒
xj ⇐⇒
U
U⇐⇒
t
i
j
i ⊆ Uto
j ⇐⇒
ij = 11 spaces
First, we
6First,
and we
M
6First,
and we
M
6 and M
whose as
whose as
whose as
If we den
ments
is
If
we den
ments
is
If
we den
ments is
an upper
pace since
an
upper
Shiraki’s
pace since
Shiraki’s
and consider the permutation σ =
,
k1 k2 · · · kn
the topogenous matrix PσT TX Pσ is upper triangular: if
i > j and tσ(i)σ(j) = 1, we would have xσ(i) < xσ(j) and
therefore
Mkj Hwhich contradicts the ordering of
Cuevas RozoM
JL,kSarria
i < Zapata
Mk , hence tσ(i)σ(j) = 0.
x15 0 1 1 1
0 •


0 0 1 0 1 1
T

.
P σ T X Pσ = 
•
x4 0 0 0 1 1 1

0Nat.041(158):127-136,
0•
1
Rev. Acad. Colomb. Cienc. Ex. Fis.
de 2017
x30 1 enero-marzo
0
0
0
0
0
1
doi:
http://dx.doi.org/10.18257/raccefyn.409
•
x
Example 0.13. Consider the topological space X, repre•x5
sented by the next Hasse diagram
and topogenous matrix:
Remark 0.14. Observe that, in general, the permutation
•
x1
σ constructed using the relations
in (3) is not unique. In
example 0.13, we could have used σ = (165)(23) (and,
in this case, no other!) to triangularize TX obtaining the
upper triangular matrix:


1 1 1 1 1 1
 0 1 0 0 1 1


 0 0 1 1 1 1
T

.
P σ T X Pσ = 

 0 0 0 1 1 1
 0 0 0 0 1 1
0 0 0 0 0 1
•x5
•x1
•x4
•x1
•x3
•x4
•x2
•x2

•x3
•x6
•x6

1 0 0 0 1 0

1 1 0 1 1 0



11 00 10 00 11 00

TX = 
11 01 00 11 11 00
1 0 1 0 1 0
0 0 0 0 1 0

TX = 
 1 0 0 1 1 0
1 1 1 1 1 1
 0 0 0 0 1 0
1 1 1 1 1 1
First, we find that: M1 = 5, M2 = M3 = 2, M4 = 3, M5 =
6 and M6 = 1. Therefore we obtain the permutation
First, we find that: M1 = 5, M2 = M3 = 2, M4 = 3, M5 =
6 and M6 = 1. Therefore
1 2 3 we4 obtain
5 6 the permutation
σ=
= (165)
6 2 3 4 1 5
1 2 3 4 5 6
σ=
= (165)
6 2 3is given
4 1 by5
whose associated matrix
opological
d topogeopological
trix to an
 is given by

d topoge- whose associated matrix
0 0 0 0 1 0
trix to an



1 0 0 0 0
0

0
0
0
1
0
0

0
0
1
0
0
0
.
Pσ = 
0
1
0
0
0
0

0 0 1 0 0
x of a T0
0

0
.
0 0
0 1
0 0
0 0
0 0
1
Pσ = 
us matrix


x of a T0
0
1 0
0 0
0 1
0 0
0 0
0

k
∈
I
;
if
0 0 0 0 0 1
n
us matrix
1 0 0 0 0 0
k = xσ(k) , the new ordering of the elek ∈ In ; if If we denote by x
ments is represented in the topogenous matrix as follows:
If we denote by x
k = xσ(k) , the new ordering of the ele(3) ments is represented in the topogenous matrix as follows:
•
x6
(3)
•
x6
·· n
,
•
x5
· · kn 

·· n
1•
gular: if,
x15 1 1 1 1
· · kn
•
x4 0 1 0 1 1 1
xσ(j) and


gular: of
if
0 0 1•
dering
x30 1 1
T

.
x=4 
Pσ TX Pσ•
xσ(j) and
0 0 0 1 1 1
•
x

2
dering of
x30 1 1
0 0 0•

•
x2 0 0 0 0 0 1
X, repre•
x1
s matrix:
X, repre

Remark 0.14. Observe that,
x
1 •
1 1in1 general,
1 1 the
1 permutation
s matrix:
σ constructed using the
not unique. In
0relations
1 0 in
1 (3)
1 is1
 have

example 0.13, we could
used
σ
=
(165)(23)
(and,


0 0 1 0 1 1
PσTno
TXother!)
Pσ = 
.
in this case,
triangularize
T
obtaining
the
to

X
 0 0 0 1 1 1
upper triangular matrix:
 0 0 0 0 1 1

0 0 0 0 0 1 
1 1 1 1 1 1
 0 1 0 0 1 1
 in general, the 
Remark 0.14. Observe that,
permutation
 0 0 1 1 1 1
T

 .unique. In
Pσusing
= relations
TX Pσthe
σ130
constructed
is
not
0 0 0in (3)

1 1 1

(and,
example 0.13, we could have
0 0used
0 σ0 =1 (165)(23)
1
in this case, no other!) to 0triangularize
T
obtaining
the
X
0 0 0 0 1
Remark 0.17
Example 0
bering its ve
2
•
For example
x5 x8 then
the edges w
then the firs
matrices are
Despite of this lack of uniqueness to choose such permutation σ, the resulting T0 spaces are always homeomorphic
(Theorem 0.6), so the topological properties are the same.
Remark 0.15. From now on, when (X, T ) is a T0 space,
we assume a fixed ordering in the elements of X such that
its topogenous matrix TX is upper triangular.
T
Stong matrix
Definition 0.16. Given X a T0 space, we define the Stong
matrix SX = [sij ] as the square matrix of size n × n that
satisfies
1 , xi xj and there is no k with xi < xk < xj
sij =
0 , in other case.
S
A simple method to calculate the topogenous matrix TX
and the Stong matrix SX of the space X, from the as- Relation
sociated Hasse diagram, is described below. Number the matrices
vertices so that xi < xj =⇒ i < j, that is, number them
from bottom to top ensuring that the topogenous matrix
We can reco
is upper triangular. For each i = j:
matrix and v
• tij = 1 if, and only if, there exists a chain whose we see that t
of the transi
minimum is xi and maximum is xj .
• sij = 1 if, and only if, (xi , xj ) is an edge of the diagram.
Remark 0.17. tij = 0 ⇒ sij = 0 and sij = 1 ⇒ tij = 1.
Example 0.18. Consider the Hasse diagram (left) numbering its vertices as described before (right):
an7 edge of•x
the
• sij = 1 if,• and only•if, (xi , xj ) is •x
8 diagram.
•x6
•
Remark 0.17. tij = 0 ⇒ sij = 0 and sij = 1 ⇒ tij = 1.
•x3
•x4 •x5
•
•
•
Example 0.18. Consider the Hasse diagram (left) numbering its vertices
as described
before •x
(right):
•x2
•
•
1
For example,• for x5 we•have x5 x5 ,•x
x57 x6 , x•x
8 x7 and
5 x5 x8 then the fifth row of TX is [0 0 0 0 1 1 1 1]. Also,
•
the edges with initial
point x1 are (x1 , x•x
3 )6 and (x1 , x4 ),
then the first row of SX is [1 0 1 1 0 0 0 0]. The associated
•x3
•x4 •x5
• are the •following:
•
matrices
t unique. In
5)(23) (and,
btaining the
•
•
•x3
•
•x4 •x5
Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 41(158):127-136, enero-marzo de 2017
doi: http://dx.doi.org/10.18257/raccefyn.409
•x2
•x1
•
•
For example, for x5 we have x5 x5 , x5 x6 , x5 x7 and
x5 x8 then the fifth row of TX is [0 0 0 0 1 1 1 1]. Also,
the edges with initial point x1 are (x1 , x3 ) and (x1 , x4 ),
then the first row of SX is [1 0 1 1 0 0 0 0]. The associated
matrices are the following:
.

ch permutameomorphic
re the same.
a T0 space,
X such that
.
ne the Stong
e n × n that
i
•x6
•
< xk < xj
s matrix TX
rom the asNumber the
umber them
nous matrix
TX
1
0

0

0
=
0

0

0
0

SX
1
0

0

0
=
0

0

0
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
1
0
1
1
1
0
0
1
1
0
1
1
1
1
0

1
1

0

1

1

1

0
1
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
0

0
0

0

0

0

1

0
1
remark
would
imply k < whose
j <k+
1, a contradiction.
For the 0.15
second
subdiagonal,
elements
are sk,k+2 ,
For
the
second
subdiagonal,
whose
elements
s57 .sk,k+2
Since,
1 k 6, there are zeros in the entries s35 andare
and
s
.
1s56 =
k
6,
there
are
zeros
in
the
entries
s
57 Since
s67 = 1 we have t56 = t67 = 1 =⇒ t3557 = 1 (theorem
s
=
1
we
have
t
=
t
=
1
=⇒
t
=
1
(theorem
s0.5)
56 =
67
56
67
57
and thus we add a one
in associated
the entry
(5,7).
In thespaces
case
Matrices
to finite
topological
0.5)
thus is
weno
add
a one in thebecause
entry (5,7).
0 there
modification,
s34 =Ins45the
= case
0:
s35 =and
because s34 = s45
s35 = 0 there is no modification,

 = 0:
1 0 1 1 0 0 0 0
01 10 01 11 00 00 00 00


0 1 0 1 0 0 0 0


0 0 1 0 0 0 0 0


0 0 1 0 0 0 0 0


0
0
0
1
0
1
0
0


(2)
(2)


= [s(2)
SX
0 0 0 1 0 1 0 0
ij ] = 

(2)
0
0
0
0
1
1
1
0


SX = [sij ] = 

0 0 0 0 1 1 1 0


0 0 0 0 0 1 1 1


0 0 0 0 0 1 1 1


0 0 0 0 0 0 1 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1
For the third subdiagonal, with elements sk,k+3 , 1 k For the third subdiagonal, with(2)
elements
sk,k+3 , 1(2) k (2)
(2)
5, we have zeros in the entries s(2)
25 , s(2)
36 , s(2)
47 and s(2)
58 . For
5, we have zeros(2)
in the(2)
entries s25 , s36 , s47 and s58 . For
example, since s46
= s67
= 1 then t46 = t67 = 1 =⇒ t47 =
(2)
(2)
example,
sincewe
s46add
= sa67one
= 1inthen
t47 =
46 = t67
1,
and hence
the tentry
(4,=
7).1 =⇒
Similarly
1, and hence we add a one(2)in the(2)entry (4, 7). Similarly
for the entry (5, 8) since s(2)
56 = s(2)
68 = 1. Entry (2, 5) is
for the entry (5, 8) since s56 = s68 = 1. Entry (2,(2)
5) is
not changed because there exists no k such that s(2)
2k =
not
(2) changed because there exists no k such that s2k =
s(2)
k5 = 1, and for the same reason we do not modify the
sentry
1, 6):
and for the same reason we do not modify the
k5 =(3,
entry (3, 6):


1 0 1 1 0 0 0 0 
10 01 10 11 00 00 00 00 




00 10 01 10 00 00 00 00 






0 00 
00 00 10 01 00 01 1


(3)
(3)


SX
=
[s
]
=


0
ij
00 00 00 10 01 11 1
(3)
(3)
1 1

SX = [sij ] = 




00 00 00 00 10 11 11 1
1






00 00 00 00 00 10 11 10 


0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1
Relation between topogenous and Stong
matrices
by the Stong matrix. For the matrix given in example 0.18 Proceeding similarly for the following subdiagonals, we
(4)
(5)
(6)
(7)


obtain the matrices SX , SX , SX and SX :
1 0 1 1 0 0 0 0
We can reconstruct thetopogenous matrix from the
Stong


0 1 0 1 0 0 0 0
 0.5,
1 0 1 1 0 0 0 0
matrix and vice versa. 
In
the
first
case,
using
theorem
0 0 1 0 0 0 0 0
0 1 0 1 0 1 0 0 

is the incidence matrix
chain whose we see that the topogenous


0 0matrix
0 1 0 1 0 0
0 0 1 0 0 0 0 0 


SX = [sclosure
of the transitive
ij ] =  of the binary relation represented



0 0 1 1 0 0
0 for0 the
0 the0 matrix
0 following
1 0 1 subdiagonals,
1 1
(4)
by the Stong matrix. For
in 1example
similarly
we


0 0 0 0 given
 0.18 Proceeding S
=
0
1
1
X
0(4)0 (5)
(6)1 1 1
(7) 1 

0
0


obtain the matricesSX , SX , SX and SX : 

10 00 10 10 00 00 01 00
0 0 0 0 0 1 1 1 


 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
1



10 00 10 10 00 00 01 00 
 0 0 1 0 0 0 0 0
00 10 00 10 00 10 00 01
 subdiwe add ones in each
subdiagonal. The first


upper
0 0 0 1 0 1 0 0



agonal,Sconsisting
0 0 1 0 0 0 0 0
X = [sij ] =of the elements sk,k+1 , 1 k 7, is


0 0 0 1 1 0 0
0
01 00 01 11 00 11 1 0 1
0if there
not modified because
is
a
k
such
that
s
=
0
(4)
k,k+1
 0 0 0 0 0 1 1 1
SX = 
0 1 0 1 0 1 1 
0




xj < xk+1 ,
and tk,k+1 = 1, therewould exist xj with xk < 
0 0 0 0 1 1 1 1



0 0 0 0 0 0 1 0
0

which is not possible since the ordering we have chosen in
00 00 01 00 00 10 1 0 1 



0 0 0 0 0 0 0 1
1
(5) 
remark 0.15 would imply k < j < k + 1, a contradiction.
00 00 00 01 00 01 1 1 0 

SX = 
0 0 0 0 1 1 1 1
Foradd
theones
second
subdiagonal,
whose elements
are subdisk,k+2 ,
0 0 0 0 0 0 0 1 

we
in each
upper subdiagonal.
The first
0 0 0 0 0 1 1 1
Since
1 k consisting
6, there are
in the entries
ks57
. 7,
is
agonal,
of zeros
the elements
sk,k+1s,351 and



10 00 10 10 00 10 01 00
= s67 = 1because
we haveif tthere
=⇒ tthat
(theorem
s56modified
56 = tis
67 a=k1such
57 =s1
=0
not
k,k+1
00 10 00 10 00 10 10 0
1
0.5)tk,k+1
and thus
addwould
a oneexist
in the
case
= 1, we
there
xj entry
with x(5,7).
xk+1
,
and


k < xIn
j <the




0
0
1
0
0
0
0
0
=
0
there
is
no
modification,
because
s
=
s
=
0:
s
35 is not possible since the ordering we have
34
45
which
chosen
in
 1 0 1 1 0 1 1 0


 k < j < k + 1, a contradiction.

0 0 0 1 0 1 1 1
(5)
remark 0.15 would imply

0 1 0 1 0 1 1 1
SX = 
1 0 1 1 0 0 0 0


For the second subdiagonal,
whose
elements
are
s
,

00 00 01 00 10 10 10 10
0 1 0 1 0 0 0 0 k,k+2



 in the entries s and s . Since
0 0 0 0 0 1 1 1
1 k 6, there are zeros

0 0 1 0 0 350 0 57
0 0 0 1 0 1 1 1
(6) 
0






S
=
0
0
0
0
0
0
1
0
X
t
=
1
(theorem
s56 = s67 = 1 we have t560 = 0t67 0= 11 =⇒

0 0 0 0 1 1 1 1
0 57
1 0 0
(2)
(2)



0 0 0 0 0 0 0 1 
[sijadd
] =a one in the entry (5,7). In the
0.5) andSthus
case


X =we
0 0 0 0 1 1 1 0
0 0 0 0 0 1 1 1



because
s
=
s
=
0:
s35 = 0 there is no modification,

0 0 0 0 0 134 1 45
10 00 10 10 00 10 11 00 
1



00 10 00 10 00 10 10 11

1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0




 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
1
0 1 0 0 1 1 0 1 0 0 0 1 01 01



 0 0 1 0 0 0 0 0
0 0 0 1 0 1 1 1
(6)

0 1 0 1 0 1 1 1


SX = 



0 0 with
131
0 0 0 0 1 1 1 1
,1k
For the (2)
third subdiagonal,
0 elements
1 0 1 sk,k+3
0 0


(2)

0 0 1 0 0 0 0 0


SX = [sij ] = 




(2)
(2)
(2)
(2)

0
0
0
0
0
1
1
1
0
0
0
0
1
1
1
0

5, we have zeros in the
 entries s25 , s36 , s47 ands58 . For
(7) 0 0 0 1 0 1 1 1
SX =
 (2) 0 0 0 0 1 1 1
0 0 0 0 0 0 1 0 
(2)

0 0 0 0 1 1 1 1
example, since s46 = s067 =
1 then t46 = t67 = 1 
=⇒ t47 =
Since we
Since
we
the topo
the matr
topo
the
the matr
The abov
Thebyabov
X
usi
X by usi
Algorith
Algorith
Input: S
Input:
space X.S
space X.











0 0

0 0
(6)
SX = 
0 0

0 0

Cuevas Rozo JL, Sarria Zapata
H
0 0
0 0

1 0
0 1

0 0

0 0
(7)
SX = 
0 0

0 0

0 0
0 0
1 0 0 0
0 1 0 1
0 0 1 1
0 0 0 1
0 0 0 0
0 0 0 0
0
1
1
1
1
0
1
0
1
0
0
0
0
0
1
1
0
1
1
1
1
0
1
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
1
0
1
1
1
0
0
0

1

1

1

0
1

1
1

0

1

1

1

0
1
1k
(2)
s58 . For
=⇒ t47 =
Similarly
y (2, 5) is
(2)
at s2k =
Output: Topogenous matrix TX = [tij ]n×n associated to
odify the Since
we reached the last upper subdiagonal, we obtain
Output:
Topogenous
matrix TX
the T0 space
X.
(7)= [tij ]n×n associated to
the
topogenous
the T0 space X.matrix TX = SX , which coincides with
the
of1,example

1.matrix
if n =Topogenous
2 do T =0.18.
S, elseTX = [tij ]n×n associated to
Output:
matrix
1.
if
n
=
1,
2
do
T
=
S,
the
T
space
X.
The
above
procedure
can
beelse
applied
T0 space
0

Output:
Topogenous
matrix
TX =to
[t any
]n×nfinite
associated
to

= 1 Tdo
tij [t=ij
sij , else
2.
if
j
i
+
1
or
s
ij
Output:
Topogenous
matrix
associated to
X
by
using
the
following
algorithm.
X =
ij ]n×n

space
X.
the
T
1 do tij = sij , else
2.
X.

1. Tif00 space
nif =j 1,
2i +
do1 Tor=sijS,=else
the

3.
for j = 3 · · · n

1. if n =for
1,Topogenous
2 do
S, matrix
else from Stong matrix
1
Algorithm:
3.
=13T
· ·=

1 do tij = sij , else
2. if nif=j 1,
2ijdo
+
1.
Tor
=·snS,
else
ij =

4.
for each zero element si,i+j−1 in the j-th

tij associated
=s sij , else
2.
if j matrix
i + 1each
orSsXijzero

Input:
Stong
==[s1element
]n×n
thej-th
T0
ijdo
4.
subdiagonal
i,i+j−1 intothe
3.
j+=13or
· · ·sn
2. upper
if jfor
ifor
ij = 1 do tij = sij , else
spaceupper
X. subdiagonal
3.
for j = 3 · · · n
5.
there
exists
r such
that sinir the
= 1j-th
=
4.
element
si,i+j−1
3.
for jfor
=if3each
· · · nzero
Output:
Topogenous
matrix
[tij ]n×n
associated
5. upper
there
r such
that
sir = 1 to=
sr,i+j−1subdiagonal
do tif
=exists
1 TX =
i,i+j−1
each zero element si,i+j−1 in the j-th
space X. for
the4.
ti,i+j−1
1 element si,i+j−1 in the j-th
r,i+j−1 dofor
4.Ts0upper
each =
zero
subdiagonal
= 0.
else
t
5. upper subdiagonal
if there
exists
r such that sir = 1 =
i,i+j−1
1. ifs n = 1, 2do
dotelse
T =tS,
else
=
0.
i,i+j−1
=
1
r,i+j−1
i,i+j−1
5.
if there exists r such that sir = 1 =
6.
endif for
5.
there= exists
r such that s = 1 =
= 11 do t = sij , else ir
2.6. sr,i+j−1
if j do
iend
+t1i,i+j−1
or
forstiji,i+j−1
else
= 1 = 0.ij
sr,i+j−1 do ti,i+j−1
7.
end for
3.7.
for
=else
3for
· · · tni,i+j−1 = 0.
endjend
for
6.
else
ti,i+j−1 = 0.
endeach
for zero element si,i+j−1 in the j-th
4.6.
7.
endfor
forforprocess,
6.
end
Reversing
the
above
we obtain an algorithm to
upper subdiagonal
Reversing
the
obtain
an algorithm
find
matrix
knowingwethe
topogenous
matrix to
of
7. the Stong
endabove
for process,
7. space.
endmatrix
for knowing the topogenous matrix of
find
the
Stong
the
5.
if there exists r such that sir = 1 =
the
Reversing
the above process, we obtain an algorithm to
sspace.
r,i+j−1 do ti,i+j−1 = 1
find
the Stong
matrixprocess,
knowingwetheobtain
topogenous
matrix to
of
Reversing
the above
an algorithm
Reversing
the
above
process,
we
obtain
an
algorithm
to
the
space.
= the
0. topogenous
elsematrix
ti,i+j−1
Algorithm:
from
find
the StongStong
matrix
knowing
topogenous matrix
matrix of
find
the StongStong
matrix
knowing
thetopogenous
topogenousmatrix
matrix of
Algorithm:
matrix
from
the space.
the
6. space.
end for
Input:
Topogenous
matrix from
TX =topogenous
[tij ]n×n associated
Algorithm:
Stong matrix
matrix to
Input:
Topogenous
the
7. T0 space
endX.for matrix TX = [tij ]n×n associated to
Algorithm:
Stong matrix from topogenous matrix
the
T0 space X.
Algorithm:
Stong matrix from topogenous matrix
Output:
Stong matrix
SX =
associated
to the
Input: Topogenous
matrix
TX[s=
[tij ]n×n
associated
to
ij ]n×n
Output:
Stong
matrix SX = [sij ]n×n associated to the
space
X.
T
space
X.
the
T
0
0
Reversing
the above process,
an algorithm
Input: Topogenous
matrix Twe
= [tij ]n×n
associatedtoto
X obtain
X.
TInput:
0 space Topogenous
[tij ]n×n associated
find
StongX.
matrix matrix
knowingTXthe= topogenous
matrix ofto
the the
T0 space
1. Tfor
i=
1X.
···n
Output:
Stong
matrix
S
=
[s
]
associated
to the
space
the
X
ij
n×n
0
the space.
1.
for i X.
= 1···n
space
T
0
Output: Stong matrix SX = [sij ]n×n associated to the
2.
forStong
j = 1 · · · n SX = [sij ]n×n associated to the
Output:
X.j = 1 ·matrix
T2.
0 space
forfor
iX.
= 1 · · · n· · n
T01.space
Algorithm:
topogenous
matrix
= 0 do
sij = tij , else
3.
if Stong
j i +matrix
1 or tijfrom
1. for i =if 1j· ·· ni + 1 or t = 0 do s = t , else
3.
ij
ij
ij
2.
for
j
=
1
·
·
·
n
1. for i = 1 · · · n
4.
if there exists k, between i + 1 and j − 1, such
2.
forifj there
= 1 · ·exists
· matrix
n
Input:
Topogenous
T=
[tijis]+
associated
to
X 0=do
n×n
4.
1 and
j − 1, such
tikifj==
t· kj
3.
j 1
tijbetween
2. thatfor
1=
·i·+
n1 or k,
ij = tij , else
the Tthat
0 space
tik X.
= 1 = tkj
3.
if j i + 1 or tij = 0 do sij = tij , else
dothere
0,1 else
1 si ij+ =
4.
if
exists
1 and
j − 1, such
ij i=+
ij
==
0 do
tij , else
3.
js
or k,
t sbetween
Output:
Stong
matrix
SX ijs=ij [s
ij1]n×n associated to the
do
s
=
0,
else
=
ij
that
t
=
1
=
t
ik there exists
kj
k, between i + 1 and j − 1, such
space end
X. if
T0 4.
5.
for
4.
if =
there
exists k, between i + 1 and j − 1, such
that
t
1
=
t
ik for
kj
5. that
end
tikdo
= s1ij==t 0, else sij = 1
1.6. for
i=
1 · · · n kj
end
for
do sij = 0, else sij = 1
6.
for
5. endend
dofor
s = 0, else sij = 1
2.
for j = 1ij· · · n
5.
end
for
132
6.
forfor
5. endend
3.
if j i + 1 or tij = 0 do sij = tij , else
6. end for
6. end for
the T0 space X.
Output: Stong matrix SX = [sij ]n×n associated to the
T0 space X.
Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 41(158):127-136, enero-marzo de 2017
doi: http://dx.doi.org/10.18257/raccefyn.409
1. for i = 1 · · · n
2.
for j = 1 · · · n
3.
if j i + 1 or tij = 0 do sij = tij , else
4.
if there exists k, between i + 1 and j − 1, such
that tik = 1 = tkj
do sij = 0, else sij = 1
5.
end for
Calculating
submodular functions
Calculating submodular functions
values
6. end for
values
Calculating submodular functions
The
entropy function
and f
Calculating
submodular
functions
values
Calculating
submodular
functions
The
entropy function
and f
values
values
The entropy function rA defined on an arbitrary FD-
The
entropy
function
and
f2014),
The
entropy
function
rA defined
an arbitrary
FDrelation
is studied
in (Sarria
et al.,on
in an attempt
relation
is
studied
in
(Sarria
et
al.,
in an
attempt
The
entropy
function
andinf2014),
to apply
the information
theory
different
structures.
The
entropy
function
andexists
to
the
information
inf
structures.
The
entropy
function
rA theory
defined
ondifferent
an
Forapply
each
N FD-relation,
there
an arbitrary
special
setFDA
For
each
N
FD-relation,
there
exists
an
special
set A
relation
is
studied
in
(Sarria
et
al.,
2014),
in
an
attempt
such
that
r
∈
λ(N
),
which
in
our
particular
case
of
A
The entropy function rA defined on an arbitrary FDsuch
that the
rAFD-relations,
∈ λ(N ), rwhich
in to
our
particular
caseFDof
The
entropy
function
defined
on
an aarbitrary
to
apply
information
theory
in
different
structures.
topological
find
A allows
relation
is studied
insubmodular
(Sarria
et al.,
2014),
inpolymatroid
an
attempt
Calculating
functions
topological
FD-relations,
allows
to
find
a
polymatroid
relation
is
studied
in
(Sarria
et
al.,
2014),
in
an
attempt
For
each
N
FD-relation,
there
exists
an
special
set
A
that
characterizes
the topological
the space,
to apply
the information
theory properties
in differentof structures.
that
characterizes
the
topological
properties
of
the
space,
to
apply
the
information
theory
in
different
structures.
such
that
r
∈
λ(N
),
which
in
our
particular
case
of
values
as
is
shown
in
(Sarria
et
al.,
2014).
Here
λ(N
)
is
the
For each NA FD-relation, there exists an special set A
as
is shown
inFD-relation,
(Sarria
et allows
al., 2014).
Here
λ(N
) is
the
For
Nr FD-relations,
there
exists
an aspecial
set
A
topological
to find
polymatroid
set
submodular
and non-decreasing
functions
such
that
suchofeach
that
A ∈ λ(N ), which in our particular case of
set
of
submodular
and
non-decreasing
functions
such
that
such
that
r
∈
λ(N
),
which
in
our
particular
case
of
that
characterizes
the
topological
properties
of
the
space,
=
N
.
We
intend
in
this
subsection
to
propose
an
N
A
f
topological
FD-relations,
allows
to
a polymatroid
The
function
and
f find
N . toWe
intend
inetthis
to
propose
an
N
topological
FD-relations,
allows
to
find
a of
polymatroid
as
shown
infind
(Sarria
al.,
2014).
Here
λ(N
is the
algorithm
the
values
ofsubsection
such
entropy
function
in
f is=entropy
that
characterizes
the topological
properties
the) space,
algorithm
to find the
values
of such
entropy
function
in
that
characterizes
the
topological
properties
of
the
space,
set
of
submodular
and
non-decreasing
functions
such
that
any
subset.
as is shown in (Sarria et al., 2014). Here λ(N ) is the
any
subset.
asf isof
in
(Sarria
etdefined
al., subsection
2014).
) isFDthe
=shown
N . We
intend
in
this
to λ(N
propose
an
N
The
entropy
function
on functions
anHere
arbitrary
A
set
submodular
and rnon-decreasing
such
that
set
of
submodular
and
non-decreasing
functions
such
that
algorithm
to
find
the
values
of
such
entropy
function
in
relation
is
studied
in
(Sarria
et
al.,
2014),
in
an
attempt
= N . that
WeEintend
propose
Nf know
, .theory
.this
. , xisubsection
X is ato
closed
set an
if,
= {xi1in
r }in⊆different
=
N
.
We
intend
in
this
subsection
to
propose
an
N
any
subset.
toWe
apply
the
information
structures.
f
algorithm
toit find
such
entropy
function
in
, . . . ,its
xof
}⊆X
is a closed
set if,
We
know
that
E =the
{xvalues
andeach
only N
if,
coincides
which
means
i1with
algorithm
toFD-relation,
find
the
values
ofirclosure
such
function
For
there
existsentropy
an
special
setthat
Ain
any
subset.
and
only
if, it coincides
with
whichthe
means
that
every
adherent
E isitsin
inclosure
E.
minimal
any
subset.
such
that
rA ∈ point
λ(N ),ofwhich
our Using
particular
case of
every
adherent
point
of
E
is
in
E.
Using
minimal
We
know
that
E
=
{x
,
.
.
.
,
x
}
⊆
X
is
a
closed
set
}
of
the
space
X,
we
have
the
following
basis
U
=
{U
i
i
x
1
r
x∈X
topological FD-relations,
allows to find a polymatroidif,
of the
space
X, we have
the
following
basis
U =if,
{Uitx }coincides
and
only
with
its closure
means
that
characterization:
x∈X
We characterizes
know that E the
= {x
⊆ X which
is aofclosed
set
if,
that
topological
the space,
i1 , . . . , xir }properties
characterization:
,
.
.
.
,
x
}
⊆
X
is
a
closed
set
We
know
that
E
=
{x
every
adherent
point
of
E
is
in
E.
Using
the
minimal
i
i
1
r
if, in
it coincides
with
closureHere
whichλ(N
means
asand
is only
shown
(Sarria et
al.,its2014).
) is that
theif,
and
with
whichthe
means
that
ofnon-decreasing
the
X,
have
following
basis
Uadherent
=if,{Uitx }coincides
x∈X
every
point
of
Espace
isitsinclosure
E.we
Using
the
minimal
set
ofonly
submodular
and
functions
such
that
every
adherent
point
of
E
is
in
E.
Using
the
minimal
characterization:
the
space
the
N=
. {U
We
intend
in⇐⇒
this
toall
propose
Nbasis
x }set
fE =
x∈X
is U
closed
ofof
Uxsubsection
∩X,
E we
= ∅have
for
xfollowing
∈
/ E an
(4)
ofX
the
space
X,
we
have
the
following
basis
Ua =
{Ufind
x }x∈X
characterization:
algorithm
to
the
values
ofx ∩
such
entropy
function
in
E
is
a
closed
set
of
X
⇐⇒
U
E
=
∅
for
all
x
∈
/
E (4)
characterization:
any subset.
Bearing
in mind
k ∅offor
thealltopogenous
E is a closed
set that
of X each
⇐⇒ column
Ux ∩ E =
x∈
/ E (4)
Bearing
in mind that
each column
topogenous
shall find
matrix represents
the minimal
open ksetofUthe
k , we
. . . ,U
xix ∩
}⊆
X
a closed
set
if,
We
that Eset=the
E know
isbyarepresents
closed
of{xXminimal
E E,
=
for
all
xshall
∈
/finding
E find
(4)
i ,⇐⇒
weE,
matrix
set∅∅is
U
one
thesetadherent
points
not
in
k , all
E only
is a one
closed
of X1with
⇐⇒
Uxropen
∩ of
E=
for
x∈
/ Ethat
(4)
and
if,
it
coincides
its
closure
which
means
one
one
the
adherent
ofcontains
E,
notthe
in
E,
finding
Bearing
in mind
that
column
k of
the by
smallest
closed
seteach
Cpoints
which
it,topogenous
using
the
every
adherent
point
ofminimal
E
is
inopen
E.contains
Using
the
minimal
the
smallest
closed
C(4),
which
using
the
matrix
represents
theset
set
Uthe
we
shall
find
characterization
given
in
by the
following
procedure:
k , it,
Bearing
in
mind
that
each
column
k
of
topogenous
}
of
the
space
X,
we
have
the
following
basis
U
=
{U
xthe
(1)
x∈X
characterization
given
in
(4),
by
the
following
procedure:
Bearing
in
mind
that
each
column
k
of
the
topogenous
one
by
one
adherent
points
of
E,
not
in
E,
finding
we
take
C
:=
E
=
{x
,
.
.
.
,
x
};
for
each
x
such
i1
k find
matrix represents
the minimal
openir set Uk , we shall
characterization:
(1)
we
take
=minimal
{x
, . . open
. , xicontains
};
xk such
matrix
set
Uk-th
weE,
shall
find
the
Cpoints
which
it,
using
the
,:=
. .adherent
.E
,the
ir set
},
we
thefor
column
of
that
k represents
∈
/C {ithe
i1consider
k ,each
1closed
one smallest
by
one
of rE,
not
in
finding
.given
, ir }, inwe
consider
the
k-th
column
of
that
k ∈
/ {ithe
one
by
one
points
of
E,
not
in
E,
finding
characterization
(4),
by
the
following
procedure:
the
topogenous
matrix
and
check
the
elements
in
rows
1 , . . adherent
the smallest
closed set C which contains it, using the
the
check
elements
ink rows
smallest
closed
set
C(4),
which
it, procedure:
using
the
we
:=matrix
E
=inand
{x
. . ,the
xcontains
for zero,
each
x
such
.topogenous
. , ir Cin(1)such
column;
they
are
all
we
take
ithe
iif
ithe
1 , . take
1 , .by
r };
characterization
given
following
. . .Ck, i(1)
inotherwise,
such
column;
ifconsider
they
are
all
zero,
we
take
iC
characterization
given
in
(4),
by
the
following
procedure:
that
/
{i
,
.
.
.
,
i
},
we
the
k-th
column
of
,
there
would
exist
j
such
that
x
is
1, =
r∈
(1)
1
r
1 all
j(4)
1
E istake
a closed
∩ E = ∅ for
x∈
/xE
we
C (1) set
:= ofEX=⇐⇒
{xiU
each
k such
1 ,x. . . , xir }; for
(2)
there
would
exist
jC1for
such
that
xrows
C
=adherent
C (1) ,Cotherwise,
we
take
:=
E
=
{x
,
.
.
.
,
x
};
each
x
such
the
topogenous
matrix
and
check
the
elements
in
=
E
∪
{x
}.
an
point
of
E
hence
we
take
j1j1 is
i
i
k
1
r
k-th column of
that k ∈
/ {i1 , . . . , ir }, we consider the (2)
= we
E
∪
{xtake
an
of
Ethat
hence
we
, .k.matrix
. ,column;
ir },
we
consider
column
of
that
/each
.topogenous
.kfor
, ir∈
in{ipoint
such
they
we
ithe
Now,
such
kif
∈
/check
{i1 ,take
. are
.the
.the
, iCrall
, k-th
j1zero,
},
consider
j1 }.
1x
1 , .adherent
and
elements
in rows
(1)each x such that k ∈
Now,
for
/check
{i1 ,exist
. .kthe
. ,of
ijrelements
,the
jsuch
we
consider
the
topogenous
and
in
rows
,
otherwise,
there
would
that
x
is
C
=
C
k-th
column
of
the
topogenous
matrix
and
check
the
k matrix
1 },
Bearing
in
mind
that
each
column
topogenous
1
j
1
zero, we take
i1 , . . . , ir in such column; if they are all
(2)and check the
the
k-th
column
of
the
topogenous
matrix
,
.
.
.
,
i
in
such
column;
if
they
are
all
zero,
we
take
ielements
=
E
∪
{x
}.
an
adherent
point
of
E
hence
we
take
C
,
.
.
.
,
i
,
j
in
such
column;
if
they
are
in
rows
i
(1)
matrix
represents
the
minimal
open
set
U
,
we
shall
find
1
r
j
1
r
1
1
k
would exist j1 such that xj1 is
C = C (1) , otherwise, there
. that
. , i(2)
, kjwould
such
ifthat
they
elements
rows
iC
,inotherwise,
xexist
is
C
Now,
each
xkadherent
such
/inwe
{iof
.E,
. . column;
, not
ijC
jsuch
},=would
we
all
zero,
we
take
,1∈
otherwise,
1 , .=
rpoints
one
byCfor
one
the
E,
finding
1,(2)
jare
1 ,exist
rthere
1in
1
E consider
∪
{x
an=adherent
point
of there
EChence
take
j1 }.
(2)
(2) would exist
,
otherwise,
there
all
zero,
we
take
C
=
C
=
E
∪
{x
}.
an
adherent
point
of
E
hence
we
take
C
the
k-th
column
of
the
topogenous
matrix
and
check
the
that
x
is
an
adherent
point
of
E,
hence
we
take
jNow,
the
smallest
closed
set
C
which
contains
it,
using
the
j
2 such
j
1
2
for each xk such that k ∈
/ {i1 , . . . , ir , j1 }, we consider
(3) for
that
xjx
adherent
point
we take
jC
Now,
each
such
that
/by
{i
, . .matrix
.of
,column;
irE,
, j1hence
}, procedure:
we
consider
,}.
. .in
. ,topogenous
i(4),
in
ifcheck
they
are
elements
in
rows
ij1an
=E
∪ {x
,isx
2 such
characterization
the
following
1such
r ,kj1∈
j21kgiven
2 the
the
of
and
the
(3) k-th column
(2)
(1)
C
=
E
∪
{x
,
x
}.
the
k-th
column
of
the
topogenous
matrix
and
check
the
,
otherwise,
there
would
exist
all
zero,
we
take
C
=
C
j1 E
we
take Cin rows
:=
.,x
for eachif xthey
ir }; column;
k such
iri,1j,1. .in
such
are
elements
ij12, . . . ,{x
,},
. . adherent
.C
, i(2)
j1otherwise,
inpoint
suchthe
column;
if they
are
in
x, .j2. .is,iC
of
E,
hence
we exist
take
jelements
i1ran
we
k-th
column
of
that
k ∈
/that
{i1rows
r ,consider
2 such
,
there
would
all
zero,
we
take
=
(2)
(3)
,
otherwise,
there
would
exist
all
zero,
we
take
C
=
C
C
=
E
∪
{x
,
x
}.
the
topogenous
and check
theofelements
that xjj12matrix
isj2an adherent
point
E, henceinwerows
take
j2 such
xj2 , is
an adherent
point
hencewe
wetake
take
. such
..=
, irEthat
in∪ {x
such
column;
if they
areofallE,zero,
i1jC,2(3)
j xj }.
(3) (1)
∪ {xj11 , xj22 }.
there would exist j1 such that xj1 is
CC=
C= E, otherwise,
Bearing in m
matrix repre
one by one
the smallest
characteriza
we take C (1
that k ∈
/ {
the topogen
i1 , . . . , ir in
C = C (1) , ot
an adherent
Now, for eac
the k-th colu
elements in
all zero, we
j2 such that
C (3) = E ∪ {
Proceeding recursively, we continue adding points to ob- Remark 0.20. In (Sarria
et al.,
2014)to is
proved
that
the
Matrices
associated
finite
topological
spaces
Remark
0.20. In (Sarria
et al.,
2014) is
proved
that
the
map
Remark 0.20. In (Sarria et al., 2014) is proved that the
such
that
=
E
∪
x
,
x
,
.
.
.
,
x
tain
a
set
C
map
j1
j2
(m)
such
= E case
∪we xcontinue
.adding
. , is
xjjm−1
tain
a=setX,C
map
in
which
the
dense
in X,that
or Remark
C (m)
j1 , xset
j2 , . E
m−1
Proceeding
recursively,
points
ob0.20. In (Sarria
et al., 2014) is proved that the
in to
the
=
X,
in
which
case
set
E
is
dense
X,
or
C (m)
(m)
(m)
−→
f(Sarria
: 2X
(m)
such
that
for Eallcase
k continue
∈
/xcontinue
{i
,
.
.
.
,
i
,
j
,
.
.
.
,
j
},
the
X,recursively,
in
which
the
set
E
is
dense
in
X,
or
C (m) ais=set
1
r
1
m−1
Proceeding
recursively,
we
adding
points
to
obRemark
0.20.
In
etZ
al.,2014)
2014)isisproved
provedthat
thatthe
the
X
Proceeding
we
adding
points
to
obRemark
0.20.
In
(Sarria
et
such
that
=
∪
,
x
,
.
.
.
,
x
tain
C
map
j{i
j.2. . , i , j jm−1
1
Zal.,
f : 2X −→
., jm−1 }, the
∈
is
such
that
for
all
k
/
,
,
.
.
C (m)
1
r
1
(m)
(m)
−→
Z
f
:
2
k-th
column
has
zeros
in
entries
i
,
.
.
.
,
i
,
j
,
.
.
.
,
j
,
in
such
that
forEEall
/xjthe
{i
,
.
.
.
,
i
,
j
,
.
.
.
,
j
},
the
C (m)aais=
1
r
1
m−1
such
that
=
∪k x∈
,
x
x
map
tain
set
Cin
1
r
1
m−1
such
that
=
∪
,
x
,
.
.
.
,
x
tain
set
C
map
I
−→
f
(I)
=
|c(I)|
j
j
j
X,
which
case
set
E
is
dense
in
X,
or
C
j
j
11
,m−1
j1 , . . . , jm−1 , in
k-th
column
has
i212 , . . . , irm−1
(m) in entries
f (I) = |c(I)|
which
case
=which
Czeros
(m)column
has
zeros
in
ijr1dense
,,dense
j.1.,..,.j.m−1
,in
jm−1
in
f : 2XII −→
X,C
in
which
case
the
X,
or
C(m)
(m)
=
X,
in
case
in
X,
or
Ck-th
is=
such
that
for ..all
kentries
∈
/the
{i1set
,iset
.1.,..E
,.E
i.r,is
,is
}, ,the
−→ fZ(I) = |c(I)|
which
case
C
=
C
X
(m)
X
(m)
(m)
.all
which
Cthat
= Czeros
f: :22 −→
−→
ZZ
iscase
such
that
forall
/find
jm−1
C is
operator
to the finite
such
for
∈
/∈
{i{i11, ,.the
Ck-th
.ri.r,,,jij1r1,,,.j.1.of
,.,.,j.{x
.m−1
, 2jm−1
in where c is the fclosure
column
has
inkkentries
i.1.,.,.,iclosure
, },
x},4 the
},the
Example
0.19.
We
shall
I −→
f (I) =associated
|c(I)|
where
c is the
closure
operator
associated
to
the satisfinite
(m) shall
,
x
}
Example
0.19.
We
find
the
closure
of
{x
2
4
k-th
column
has
zeros
in
entries
i
,
.
.
.
,
i
,
j
,
.
.
.
,
j
,
in
topological
space
(X,
T
),
is
a
polymatroid
which
,
.
,
i
,
j
,
.
,
j
,
in
k-th
column
has
zeros
in
entries
i
which
case
C
=
C
.
where
c
is
the
closure
operator
associated
to
the finite
1
r
1
m−1
1
r 1 of
the topological
X whose
topogenous
matrix
−→
(I)
|c(I)|
, xis4 }the
in topological space (X,IIT−→
Example
0.19.space
We
shall
find the
closure
{x2m−1
fisf(I)
==|c(I)|
),
a
polymatroid
which
satis(m) X whose topogenous matrix
(m)
the
topological
space
is
the
which
caseCC==C
C . . X whose topogenous matrix is the topological
fies f ∈c λ(N
hence,
the above
algorithm
which
case
space
(X, Tinoperator
),particular,
is a polymatroid
which
T ) closure
following:
the topological
space
is the
associated
to
the satisfinite
Example
find the closure of {x2 , x4 } in where
fies
f ∈toλ(N
infunction
particular,
theofabove
algorithm
T ) hence,
following: 0.19. We shall


allows
determine
the
values
f
completely.
fies
f
∈
λ(N
)
hence,
in
particular,
the
above
algorithm
T
where
c
is
the
closure
operator
associated
to
the
finite
following:
where
c
is
the
closure
operator
associated
to
the
finite
topological
space
(X,
T
),
is
a
polymatroid
which
satis,
x
}
in
Example
0.19.
We
shall
find
the
closure
of
{x
,
x
}
in
Example
0.19.
We
shall
find
the
closure
of
{x
the topological space X
matrix
4 the allows to determine the function values of f completely.
1 whose
0 1 topogenous
0 0
22 4is


allows
determine
the
values
f which
completely.
topological
space
(X,
),particular,
polymatroid
which
satis0
1
0
0
space
(X,
TTin
),function
isis aa polymatroid
satisfies
f ∈toλ(N
theofabove
algorithm
1
 matrix
T ) hence,
the topological
topological space
space XX
whose
topogenous
matrix isis the
the topological
the
whose
topogenous
following:
0
1
1
1
0
1
0
0


fies
f
∈
λ(N
)
hence,
in
particular,
the
above
algorithm
0
1
1
1
0


fies
f
∈
λ(N
)
hence,
in
particular,
the
above
algorithm
allows
to
determine
the
function
values
of
f
completely.
=
|c(K)|
be
the
cardinal
number
of
the
cloNow,
let
n
T
T


K
following:
following:
0
1
0
0
TX = 0
.
1
1
= (Sarria
|c(K)|
be
the
cardinal
number
of the
cloNow,oflet
ndetermine
1
0
1
0
0

K In

allows
to
the
function
values
of
f
completely.
0
0
1
0
0
TX =
.

allows
to
determine
the
function
values
of
f
completely.
sure
K.
et
al.,
2014)
it
is
shown
that
if
=
|c(K)|
be
the
cardinal
number
of
the
cloNow,
let
n


K


1
0
TX = 

1
0
1
0
0
sure
of
K.
In
that
if
01
0 00
1 10
1 00
10
.
n
nK (Sarria et al., 2014) it is shown
X


0
0
0
1
0


=
2
−
2
,
the
entropy
function
r
:
2
−→
R
satF
sure
of
K.
In
(Sarria
et
al.,
2014)
it
is
shown
that
if
K
A


0
n nK n
X of the cloNow,
let
=K ,|c(K)|
be the function
cardinal rnumber
1
0

0 10
1
0
=
2
−
2
the
entropy
:
2
−→
R
satF
0 10
0 01
TX =
K
A
n
n
X
K
0 1
0
0 11
0
1 .
00
isfies
=
2
−
2
,
the
entropy
function
r
:
2
−→
R
satF
K
A
=
|c(K)|
be
the
cardinal
number
of
the
cloNow,
let
n
= |c(K)|
be et
theal.,
cardinal
of the
clo-if
Now,
let K.
nKK In
sure
(Sarria
2014) number
it is shown
that
00



1 }.00
0Since
=
isfies of
(1)
00
10
01
0 {x
0,x
0
1
0
TTXX
Initially, we take
C=
=
. . the first column sure
n
nK (Sarria et al., 2014) it is shown
X
2
4
isfies
(1)
sure
of2K.
K.
In
that
of
In
(Sarria
et
al.,
2014)
it
is
shown
that
ifif
=
−
2
,
the
entropy
function
r
:
2
−→
R
satF

K
A

=
{x
,
x
}.
Since
the
first
column
Initially,
we
take
C
1
0
0
0
0
1
0
0
0
0
0
1
2
4
notcolumn
added FF
has
zeroswe
in take
rowsC2(1)and
4, 2the
element
= {x
, x4 }.
Since x
the
Initially,
nn − 2
nn
XX −→ R sat1 isfirst
KK , the entropy function r
=
2
:
2
=
2
−
2
,
the
entropy
function
r
:
2
−→
R
satisfies
KK
not added
has
in rows
and
0the
2SI AA
(1)
00 4,
00 aelement
00one11 in x
1 issecond
; the
thirdC2
the
row isfies
to
Czeros
(1)
notcolumn
added
has
zeros
in take
rows
2column
and
4,0 has
the
element
x
1 isfirst
rA (I) = ln |A| − 2SI ln 2
(1) we
= {x
Since
the
Initially,
isfies
2 , x4 }.
;
the
third
column
has
a
one
in
the
second
row
to
C
(2)
(1)
|A|I ln 2
r (I) = ln |A| − 2S
(1)= {x2 ,has
x}.
Every
time
we
add
a
and
so we
make
C
(1)
; we
the
third
aelement
one
inthe
the
second
row
to
Czeros
4 }.
{x
,x3x4,,4}.
Since
the
column
Initially,
we
take
=={x
{x
Since
first
column
Initially,
take
CC(2)
isfirst
not
added
has
in
rows
2column
and
4,22,2,x
the
x
rA
A (I) = ln |A| − |A| ln 2
1time
=
x
x
}.
Every
we
add
a
and
so
we
make
C
3
4
(2)
|A|I
2S
point,
must
look
again
elements
that
had
(1)we
= {x
,the
x
,x
Every
time
wewe
add
a
and
so
we
make
C22column
3those
4 }.
not
added
has
zeros
in
rows
and
the
not
added
has
zeros
in
rows
and
4,4,2at
element
; the
third
has
aelement
one
inxx1the
second
row
to
C
1 isis
rA (I) = ln |A| − 2S ln 2
point,
we
must
look
again
at
those
elements
that
we
had
(1)we must
2S
discarded
in
the
previous
step,
in
this
case
x
.
The
first
(1)
(2)
point,
look
again
at
those
elements
that
we
had
where
II
1
; the
the
third
column
a4one
one
the
second
row
to
third
column
has
inin
the
second
row
to
CCso;we
= {x2step,
,has
x3 , axin
}.
Every
time
we
add
a where
and
make
Cprevious
(I)==lnln|A|
|A|−− |A|
lnln22
rrAA(I)
discarded
in
the
this
case
xis
Theadded
first
1 . not
(2)rows 2,
column
has
zeros
in
3
and
4,
then
x
(2)
|A|
discarded
in
the
previous
step,
in
this
case
x
.
The
first
|A|
1
where
1
=
{x
,
x
,
x
}.
Every
time
we
add
a
and
so
we
make
C
=
{x
,
x
,
x
}.
Every
time
we
add
a
and
so
we
make
C
point,
we
must
look
again
at
those
elements
that
we
had
2
3
4
2
3
4
added
column
zerosthe
in fifth
rows column
2, 3 andhas
4, then
x1inis2,not
; has
finally
zeros
3we
and
4,
to C (2)
column
has
in again
rows
2,
and
4,
then
x1x
is
added
point,
we
must
look
againcolumn
at3those
those
elements
that
we
had
(2)
point,
we
must
look
at
elements
that
had
discarded
inzeros
the
previous
step,
inhas
this
case
. not
The
first
12,
;
finally
the
fifth
zeros
in
3
and
4, where
to
C
(2)
(2)
is
not
added
to
C
,
and
as
we
have
exhausted
all
so
x
FK
SI = ;
finally
the
fifth
column
has
zeros
in
2,
3
and
4, where
to
C
5
discarded
in
the
previous
step,
in
this
case
x
.
The
first
(2)
discarded
in
the
previous
step,
in
this
case
x
.
The
first
is
not
added
column
has
zeros
in
rows
2,
3
and
4,
then
x
where
1
1
1
added
to
C (2) , and
as the
we have
exhausted
all}
so
x5(2)isofnot
SI = FK
,
x
points
space,
we
conclude
that
closure
of
{x
I⊆c(K)
is
not
added
to
C
,
and
as
we
have
exhausted
all
so
x
2
4
=
F
S
5
I
is
not
added
column
has
zeros
in
rows
2,
3
and
4,
then
x
not
added
column
has
zerosthe
in rows
2,
3 and
4,the
then
x11inis 2,
; finally
fifth
column
zeros
and
4,
to C of
K
points
space,
we
conclude
thathas
closure
of 3{x
I⊆c(K)
(2)
2 , x4 }
(2)of
is
C
=
C
=
{x
,
x
,
x
}.
(2)
(2)
,
x
}
points
space,
we
conclude
that
the
closure
of
{x
I⊆c(K)
2
3
4
2
4
;
finally
the
fifth
column
has
zeros
in
2,
3
and
4,
to
C
(2) added
; Cfinally
the fifth
zeros
2, 3 and 4,
to
not
to C4column
as we
haveinexhausted
all
so C
SI = FK
is
Cx5=is
}. , and has
(2) = {x2 , x3 , x(2)
is
C
=
{xwe
xto
x4(2)
}., ,and
FFF
2 ,to
3 ,C
not
added
C
and
aswe
wehave
have
exhausted
all
so
= I⊆c(K)
SIII==
isisof
not
added
as
exhausted
so
xCx55=
points
space,
conclude
that
the
closure
of {x2 , xall
SM
K
4}
KK
MI = F
The
above
comments
help
find
thethe
closure
of an
arbitrary
K
(2)
,
x
}
points
of
space,
we
conclude
that
the
closure
of
{x
,
x
}
points
of
space,
we
conclude
that
closure
of
{x
I⊆c(K)
is
C
=
C
=
{x
,
x
,
x
}.
M
=
Ic(K) FK
22 44
2
3 help
4 find the closure of an arbitrary
I I⊆c(K)
The
above
comments
(2)comments
using
the
following
algorithm.
Ic(K)
(2)
The
above
help
find
the
closure
of
an
arbitrary
is
C
=
C
=
{x
,
x
,
x
}.
issubset
C
=
C
=
{x
,
x
,
x
}.
33 44 algorithm.
MI = Ic(K)
subset using the22following
FK
subset
usingcomments
the following
The above
help algorithm.
find the closure of an arbitrary
|A|
I +
I)
MII=
M
==2(M
FFS
Ic(K)
KK
|A| = 2(MI + S
I)
Theabove
above
comments
helpfind
findthe
the closureofofan
anarbitrary
arbitrary
The
comments
help
subset
using
the
following
|A| = 2(M
I + SI )
Ic(K)
Algorithm:
Closure
of aalgorithm.
subsetclosure
I⊆X
Ic(K)
subsetusing
usingthe
the
following
Algorithm:
Closure
of algorithm.
aalgorithm.
subset I ⊆ X
subset
following
|A| M
= I2(M
+ SI )
Algorithm: Closure of a subset I ⊆ X
as Ifollows:
We can rewrite SI and
M
asIIfollows:
Input: Topogenous matrix TX = [tij ]n×n associated to We can rewrite SI and
I2(M
|A|
=
+SSII))
|A|
=
2(M
+
Algorithm:
Closure of a subset
⊆]n×n
X associated to We can rewrite
I as follows:
Input:
[tij
SI and M
X = I
the
spaceTopogenous
X, a subsetmatrix
I ⊆ X.T
Input:
Topogenous
matrix
T
=
[t
]
associated
to
n
X
ij
n×n
SI =rewrite
FK = M
(2n − 2nnK
)
Algorithm:
subsetII⊆⊆XX
Algorithm:
subset
the space X, Closure
aClosure
subset of
Iof
⊆aaX.
K
I as(2follows:
SI = SFIKand
the
spaceTopogenous
X, a subsetmatrix
I ⊆ X.TX = [tij ]n×n associated to We can
= n − 2n K )
Input:
I⊆c(K)
I⊆c(K)
Output: Closure of I: c(I) = C.
SI rewrite
=rewrite
F
=
(2
−
2
)
and
M
as
follows:
Wecan
can
S
K
and
M
as
follows:
We
S
II
II
I⊆c(K)
I⊆c(K)
Output:
Closure
of matrix
I:
n nK
Input:
Topogenous
matrix
TXXC.
]n×n associated
associated to
to
Input:
Topogenous
T=
the
space
X,
a subset
I c(I)
⊆ X.
I⊆c(K)
I⊆c(K)
Output:
Closure
of I:
c(I)
=
C.== [t[tijij]n×n
n
n
SI = (2 − 2 2n)K −n
=: 2nn SI∗∗
FK = 2
1.space
Define
Caa=
I andIIr⊆⊆
=X.
1.
nnn1 −
−n
the
space
X,
subset
X.
K
the
X,
subset
n
nK2
Kn −n =: 2n S ∗
1
−
=
2
1.
Define
C
=
I
and
r
=
1.
n
S
=
F
=
(2
−
2
)
K
SII= I⊆c(K)FKK== I⊆c(K)
(2 −12− 2)
Output:
=: 2 SII
2 I⊆c(K)
1. DefineClosure
C = I of
andI:rc(I)
= 1.= C.
2.
while
r
=
1
I⊆c(K)
I⊆c(K)
I⊆c(K)
Output:
Closure
of
I:
c(I)
=
C.
I⊆c(K)
I⊆c(K)
Output:
Closure
n −n
n ∗
2.
while
r = 1 of I: c(I) = C.
= 2n I⊆c(K)
1.
1 − 2 K =: 2 SI
2. Define
while rC==1I and r = 1.
n
n
−n
n
n
−n
n
K
K
3.
for
x
∈
/
C
i
−22
=:22nSSI∗I∗
=:
==22 Define
=
andrr==1.1.
I⊆c(K) 11−
1.1.
3. Define
forCrC
xi=
∈
/1IICand
n
nK
2.
3. while
for x=
∈
/
C
M
=
F
=
(2
−
2
)
i
I⊆c(K)
I
K
I⊆c(K)
= 0 do C ← C ∪ {xi } and go to
2. while
whilerif
r==
MI = FK = (2nn − 2nnK
2.4.
11xk ∈C ttik
K)
4.
if
M
=
)
ik = 0 do C ← C ∪ {xi } and go to
Ic(K) FK = Ic(K)(2 − 2
3.
for
x
∈
/
C
I
x
∈C
i
k
step
3.
4.
if
xk ∈C tik = 0 do C ← C ∪ {xi } and go to
Ic(K)
Ic(K)
3.xxi i∈
n
nKn −n for
/CC
Ic(K)
Ic(K)
3.3. step
for
/∈
n
M
=
F
=
(2
−
2
step
3.
I
K = 2
=: 2nn MI∗∗
2Kn)K
C do
←C
∪ {x
4.
if ik =
5.
if
i=xkmax
: x0j do
∈
/ C}
r=
0. i } and go to
n nn1 −
∈C t{j
K −n nn
K
1
−
2
=: 2n MI∗
2
M
=
F
=
(2
−
2
)
M
=
F
=
(2
−
2
)
n
n
−n
I
K
K
Ic(K)
Ic(K)
I
K = 2 Ic(K) 1 − 2
i =x max
:x
∈
/ C}
do
r ∪=
0.i }i }and
t{j
=
do
←CC
∪{x
{x
goto
to
4. step 3.ifif
00jjdo
CC←
4.5.
=: 2 MI
ik =
∈Ctik
5.
if i =
max
{j
:x
∈
/ C}
do
r=
0. andgo
xkk∈C
Ic(K)
6. step
end
for
Ic(K)
Ic(K)
Ic(K)
Ic(K)
n Ic(K)
nK −n
n
∗
step
3.
3.
=2 6.
endif for
1 − 2
5.
i = max {j : xj ∈
/ C} do r = 0.
=: 2nn M∗I∗
6.
end for
n
n
−n
n
n
−n
K
K
while
=:22 M
MII
11−−22
=:
==22 ∗
5. end
max{j
{j: :xxj j∈
/C}
C}do
dorr==0.0.
5.7.
ifif ii==max
/∈
so that |A| = 2n+1
(MI∗∗ + Ic(K)
SI∗ ) and hence
7.
end while
6.
for
7. endend
while
Ic(K)
Ic(K)
so that |A| = 2n+1
(M
+
S
)
and
hence
so that |A| = 2n+1 (MII∗ + SII∗ ) and hence
endfor
for
6.6.
end
7. end while
∗
∗
so that |A| = 2n+1 (M
I I+ SI ) and hence
2S
endwhile
while
7.7. end
n+1
∗
n+1
∗
∗
rA (I)
===
ln22|A| −
sothat
that
|A|
(MII ++ln
andhence
hence
so
|A|
(M
SS2∗))and
|A| II
2n+1 SI∗
= ln(2n+1 (MI∗ + SI∗ )) − n+1
ln 2
2
(MI∗ + SI∗ )
SI∗
ln 2
= (n + 1) ln 2 + ln(MI∗ + SI∗ ) −
∗
(MI + SI∗ )
M∗
= ln(MI∗ + SI∗ ) + n + ∗ I ∗ ln 2
MI + SI
continue
Rev.
Acad. Colomb.
Cienc. Ex. Fis.we
Nat.
41(158):127-136,
enero-marzo
de 2017
Proceeding
recursively,
adding
ob(m)
suchto
= E ∪we
. , xjm−1points
tain
a set C
Proceeding
recursively,
points
tothat
obj1 , xj2 , . .adding
doi:
http://dx.doi.org/10.18257/raccefyn.409
(m)
xcontinue
Having the algorithm to find the closure of a subset133
K,
and therefore to calculate the values nK , SI∗ and MI∗ , we
can evaluate the function rA by the following algorithm.
where ∆
satisfies
for each
a subset
where B
bounds
These fu
topology
and therefore to calculate the∗ values
n , S ∗S ∗and M ∗ , we
= (n + 1) ln 2 + ln(M + SI∗ ) −K ∗I I ∗ lnI 2
following
can evaluate the function rA Iby the
(MI +algorithm.
SI )
∗
M
= ln(MI∗ + SI∗ ) + n + ∗ I ∗ ln 2
Cuevas Rozo JL, Sarria
Zapata H
MI + SI
These functions fU and fD are important
to describe the
= |X|they
− |Bcharacterize
|
fU (B)since
topology of the space,
the order
(B) = |X| − |B |
relation given infD(2):
Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 41(158):127-136, enero-marzo de 2017
doi: http://dx.doi.org/10.18257/raccefyn.409
xj denote
⇐⇒ fDthe
(xi ) sets
= fDof(xupper
(5)
xi B
where B and
i , xj ) and lower
bounds
of
B
in
(X,
),
respectively.
Algorithm:
Entropy function rA value in a subset I ⊆
Having
the algorithm
to find the closure of a subset K,
xi xj ⇐⇒ fU (xj ) = fU (xi , xj )
(6)
X
and therefore to calculate the values nK , SI∗ and MI∗ , we
functions
fU Given
and fDa are
important
to describe
algorithm.to These
can
evaluate
the function
rA by
Definition
0.21.
finite
topological
space X,the
we
Input:
Topogenous
matrix
TXthe
= following
[tij ]n×n associated
topology
of
the
space,
since
they
characterize
the
order
define
the
matrix
U
=
[u
]
associated
to
the
function
X
ij
the space X, a subset I ⊆ X.
relation
given
in which
(2): satisfies u = f (x , x ). Analomatrix
fU , as the
ij
U
i
j
Output: Value rA (I).
gously, we define the matrix DX = [dij ] associated to the
(5)
xi xj ⇐⇒ fD (xi ) = fD (xi , xj )
function fD as
the matrix satisfying dij = fD (xi , xj ).
1. Define MI∗ = SI∗ = 0.
Algorithm:Entropy function rA value in a subset I ⊆
xj ⇐⇒
fU is
(xjthe
) =topogenous
fU (xi , xj ) matrix asso(6)
xi 0.22.
Proposition
If TX
X 2. for K ⊆ X
ciated to the finite topological space X, then
space X, we
= |c(K)|.
3.
Calculate nKmatrix
Input:
Topogenous
TX = [tij ]n×n associated to Definition 0.21. Given a finite topological
T
U
=
n1
−
T
T
define
the
matrix
U
=
[u
]
associated
to
the
function
X
X
X
ij
X
the4.spaceifX,
subsetdo
I⊆
← SI∗ + 1 − 2nK −n
I ⊆a c(K)
SI∗X.
fU , as the matrix which satisfies uij = fU (xi , xj ). AnaloT
TX
T[dXij ] associated to the
Output:else
Value
X = n1D−
gously, we define theDmatrix
X =
MI∗rA
←(I).
MI∗ + 1 − 2nK −n
function
f
as
the
matrix
satisfying
dijsize
= fnD×
(xni , such
xj ). that
where 1 =D[aij ] is the square matrix of
1.5. Define
MI∗ = SI∗ = 0.
end for
i, j ∈ If
InT
. X is the topogenous matrix assoaij = 1 for all0.22.
Proposition
2.6. for
K ⊆ Xr (I) = ln(M ∗ + S ∗ ) + n + MI∗
ln
2
Calculate
A
ciated to the finite topological space X, then I
I
MI∗ +SI∗
n
Proof. If E = {xi , xj } , we show that |E| = k=1 tik tjk :
3.
Calculate nK = |c(K)|.
T
X = n1 − TX TX
in fact, we have theUequivalences:
4.
if I ⊆ c(K) do SI∗ ← SI∗ + 1 − 2nK −n
T
Functions f∗U and∗ fD : matrices
UX and DX
DX
= n1
−T
⇐⇒
xi xkX TyXxj xk
xk ∈ E
else MI ← MI + 1 − 2nK −n
⇐⇒ tikmatrix
= 1 y of
tjksize
= 1n × n such that
where 1 = [aij ] is the square
5. end
Let
(X, Tfor
) be a finite topological space, U its minimal ba=
1
for
all
i,
j
∈
I
.
a
ij
n
⇐⇒
t
t
=
1
op
ik jk
sis and D the minimal basis ∗for X∗ , the opposite
MI∗ space of
ln 2
6. Calculate r (I) = ln(MI + SI ) + n + M ∗ +S
∗
n
X, and considerA the non-decreasing
submodular
I
Ifunction
then each
provides
a one in the sum nk=1ttikikttjk
k ∈
jk:,
Proof.
If Ex=
{xE
i , xj } , we show that |E| =
k=1
f∆ : 2X −→ Z defined by
T
thus
having
the
required
equality.
Now,
if
T
T
=
[c
X
ij ]
in
fact,
we have
the
equivalences:
X
n
then
c
=
t
t
=
|E|
and
hence
ij
k=1 ik jk
Functions fU and
fD:=
: matrices
xk ∈ E ⇐⇒ xi xk y xj xk
f∆ (I)
qJ (I) UX and DX
T
]ij .
uij = fU (xi , xj ) = |X| − |E| = n − cij = [n1 − TX TX
J∈∆
⇐⇒ tik = 1 y tjk = 1
Let (X, T ) be a finite topological space, U its minimal ba⇐⇒ tik tjk = 1
sis and D the minimal basis for X op , the opposite space of
Let A = {xi , xj } . By a similar argument above,
we have
X, and consider the non-decreasing submodular function
nx ∈ E provides a one in the sum n t t ,
then
each
k
|A| = k=1 tki tkj from which we get
k=1 ik jk
f∆ : 2X −→ Z defined by
T
thus havingthe required equality. Now, if TX TX
= [cij ]
n
T
=
t
t
=
|E|
and
hence
then
c
ij
ik
jk
dij = fD (x
k=1
i , xj ) = |X| − |A| = [n1 − TX TX ]ij . f∆ (I) :=
qJ (I)
T
, xj ) =For
|X|the
− |E|
= Tn −space
cij = [n1 − TX TX
]ij .
u
ij = fU (xi0.23.
J∈∆
Example
next
0
ln 2
SI∗ )
ln 2
ubset K,
d MI∗ , we
gorithm.
ubset I ⊆
ciated to
These functions fU and fD are important to describe the
topology of the space, since they characterize the order
relation given in (2):
134
xi xj ⇐⇒ fD (xi ) = fD (xi , xj )
(5)
xi xj ⇐⇒ fU (xj ) = fU (xi , xj )
(6)
Definition 0.21. Given a finite topological space X, we
define the matrix UX = [uij ] associated to the function
xr ∈
•x5
•x3
fU (B) = |X| − |B |
fD (B) = |X| − |B |
where B and B denote the sets of upper and lower
bounds of B in (X, ), respectively.
xr ∈
•x6
where ∆ ⊆ 2X and qJ : 2X → {0, 1} is the map which
satisfies
1, IJ
qJ (I) :=
0, I⊆J
for each I, J ⊆ X. In (Abril, 2015) is proved that if B is
a subset of X then
Proof.
•x4

•x2
0
1
0
0
0
0
0
1
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
0
matrices UX and DX are:

2 3
3 1

6 5
UX = 
3 3

4 4
5 5
6
5
5
6
6
6
3
3
6
3
4
5
4
4
6
4
4
5
TX
1
0

0
=
0

0
0
•x1

1
1

0

1

1
1

5
5

6

5

5
5
Therefore
|M | =
r
Proposition
DX from th
reverse proc
UX and from
equivalent t
t
Therefore, w
using the fo
0
3
TX = 

06

UX =
03

04
05


16


15

15
Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 41(158):127-136,
enero-marzo de 2017
5 5 6 5 5 5
doi: http://dx.doi.org/10.18257/raccefyn.409
matrices UX and DX are:


25 36 66 35 45 55
36 15 55 35 45 55


66 55 54 65 65 65



UDXX==

35 35 65 33 43 53
45 45 65 43 42 52
55 55 65 53 52 51
15
05
06
06
03
16
03
04
04
16
14
04


5 X6 be6a finite
5 5 topological
5
Let
space and

6
5
5
5
5
5
a
subset
of
X,
then


6 5 4 5 5 5

DX = 
m3 3
5 5 5n 3 

fU (M )=
5 n5− 5 3 2 tik2r
5 5 r=1
5 3 k=1
2 1
n
m
Proposition 0.24.
Let
X
be
a
finite
topological
space and
fD (M ) = n −
trik
M = {xi1 , . . . , xim } a subset of
X,
then
r=1 k=1
Proposition 0.24.
M = {xi1 , . . . , xim }
fU (M ) = n −
fD (M ) = n −
ove, we have
X ]ij .
01
05
03
04
n
r=1
n
r=1
m
ti k r
trik
k=1
m
k=1
k=1
equivalent r=1
to having
r=1
k=1
tij = 1 ⇐⇒ ujj = uij ⇐⇒ dii = dij
Matrices
associated
tomatrices
finite topological
and
Proposition
0.22
to obtain
UX spaces
Therefore, we
canallows
reconstruct
the the
topogenous
matrix
by
the
topogenous
matrix
T
.
Consider
now
the
Dusing
X from
X
the following algorithms:
reverse process of obtaining the topogenous matrix from
UX and from DX . In our matrix language, (5) and (6) are
equivalent
to having
Algorithm:
Topogenous matrix from UX
tij = 1UX
⇐⇒
uij associated
⇐⇒ dii = to
dijthe function
Input: Matrix
=u
[ujjij ]=
n×n
fU of X.
Therefore, we can reconstruct the topogenous matrix by
Output:
Topogenous
matrix TX = [tij ]n×n associated to
using
the following
algorithms:
the space X.
1. for i = 1 · · · n
Algorithm: Topogenous matrix from UX
2.
for j = 1 · · · n
Input: Matrix UX = [uij ]n×n associated to the function
fU 3.of X. if i > j do tij = 0, else
4.
tij = δ(uij matrix
− ujj , 0)T = [t ]
Output:
Topogenous
associated to
X
ij n×n
the5.spaceend
X. for
1.6. for
1···n
endi =
for
for j = 1 · · · n
2.
3.
if i > j do tij = 0, else
4.
tij = δ(uij − ujj , 0)
Conflict
5.
end for
no confli
Algorithm: Topogenous matrix from DX
6. end for
Referen
Input: Matrix DX = [dij ]n×n associated to the function
fD of X.
Abril J
Output: Topogenous matrix TX = [tij ]n×n associated to homotop´
ciones su
the space X.
sidad Na
1. for i = 1 · · · n
gotá, Co
2.
for j = 1 · · · n
Alexand
(N.S.)2,
3.
if i > j do tij = 0, else
Proof.
xr ∈ M ⇐⇒ xik xr for all k = 1, . . . , m
⇐⇒ tik r = 1 for all k = 1, . . . , m
m
⇐⇒
t ik r = 1
k=1
4.
xr ∈ M ⇐⇒ xr xik for all k = 1, . . . , m
⇐⇒ trik = 1 for all k = 1, . . . , m
m
⇐⇒
trik = 1
5.
Cuevas
matrices
(Trabajo
Colombi
tij = δ(dij − dii , 0)
end for
6. end for
k=1
Remark 0.25. From proposition 0.24 and above algorithms, we see that if M = {xi1 , . . . , xim } ⊆ X we have:
Therefore
|M | =
n
r=1
m
k=1
ti k r
and |M | =
n
r=1
m
k=1
trik
.
Proposition 0.22 allows to obtain the matrices UX and
DX from the topogenous matrix TX . Consider now the
reverse process of obtaining the topogenous matrix from
UX and from DX . In our matrix language, (5) and (6) are
equivalent to having
tij = 1 ⇐⇒ ujj = uij ⇐⇒ dii = dij
Therefore, we can reconstruct the topogenous matrix by
using the following algorithms:
Algorithm: Topogenous matrix from UX
Input: Matrix UX = [uij ]n×n associated to the function
f of X.
fU (M ) = n −
n
fD (M ) = n −
n
r=1
r=1
m
k=1
m
k=1
δ(fU (xr ) − fU (xik , xr ), 0)
δ(fD (xik ) − fD (xik , xr ), 0)
Therefore, functions fU and fD can be calculated for any
subset M of the space X, from the functions values in
the subsets of cardinality one and two through the above
explicit formulas.
Conclusions
135
Krishna
on a fini
Sarria,
entre los
funcione
31-50.
Shiraki,
Fac. Sci
Stong R
Amer. M
Therefore,
functions
fU and
fD can
calculated
for any
subset
M of
the space
X, from
thebefunctions
values
in
fD of X.
subset
M
of
the
space
X,
from
the
functions
values
in
the subsets of cardinality one and two through the above
[tijthrough
]n×n associated
to
Output:
Topogenous
matrix
TX =
the
subsets
of cardinality
one and
two
the above
explicit
formulas.
the
space
X.
explicit
formulas.
Cuevas Rozo
JL, Sarria Zapata H
1. for i = 1 · · · n
Conclusions
2.
for j = 1 · · · n
Conclusions
3.
if i > j do tij = 0, else
We have seen how the considered matrices make easier the
We4.have seen
the
matrices
make
easier the
= δ(d
− dspaces
tijhow
topological
study
in
finite
we use
submodular
ij considered
ii , 0) when
topological
study in finite
spaces when
we use
functions, achieving
to eliminate
the need
to submodular
draw Hasse
5.
end for
functions,
to eliminate
the needinto
draw Hasse
diagrams toachieving
find its values
as was worked
(Abril,
2015)
diagrams
to
find
its
values
as
was
worked
in
(Abril,
2015)
6.
end
for
and (Sarria et al., 2014). In future works, it can
be
and
(Sarria
et
al.,
2014).
In
future
works,
it
can
be
studied complexity of the shown algorithms and try to find
studiedtopological
complexityconcepts
of the shown
algorithms
try to find
other
which
could beand
characterized
other
topological
concepts
which could
from
this
matrix
point
of view.
Remark
0.25. From
proposition
0.24 be
andcharacterized
above algofrom
this
matrix
point
of
view.
rithms, we
that if The
M =authors
{xi1 , . . declare
. , xim } that
⊆ X they
we have:
Conflict
of see
interest.
have
Conflict
no conflictofofinterest.
interest. The authors declare that they have
no conflict of interest.
m
n
References
function References
δ(fU (xr ) − fU (xik , xr ), 0)
fU (M ) = n −
function
r=1
k=1
Abril J. A. (2015). Una aproximación a la noción de
J. A.
(2015).
Una
aproximación
la noción
de
homotopı́a
entre
espacios
topológicos
finitosa desde
las funciated to Abril
entre
espacios
topológicos
finitos
desde
las
funciated to homotopı́a
ciones submodulares (Trabajo
Final de Maestrı́a). Univerm
ciones
submodulares
de Maestrı́a).
Univern (Trabajo
Final
sidad Nacional
de Colombia.
Facultad
de Ciencias.
Bosidad
Nacional
Facultad
Ciencias.
(M ) = n de
− Colombia.
δ(fD
(xik ) − fde
D (x
ik , xr ), 0) Bogotá,fDColombia.
gotá, Colombia. r=1 k=1
Alexandroff P. (1937). Diskrete Räume. Mat. Sb.
Alexandroff
P. (1937). Diskrete Räume. Mat. Sb.
(N.S.)2,
501-518.
Therefore,
functions fU and fD can be calculated for any
(N.S.)2,
501-518.
subset M
of the
X, from
the functions
values yin
Cuevas
Rozo
J. space
L. (2016).
Funciones
submodulares
Cuevas
Rozo
J. L. (2016).
Funciones
submodulares
y
the subsets
of estudio
cardinality
oneespacios
and two topológicos
through
thefinitos
above
matrices
en el
de los
matrices
en
el
estudio
de
los
espacios
topológicos
finitos
explicit
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(Trabajo Final de Maestrı́a). Universidad Nacional de
(Trabajo
de Maestrı́a).
Nacional de
Colombia.Final
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de Ciencias.Universidad
Bogotá, Colombia.
Colombia. Facultad de Ciencias. Bogotá, Colombia.
Krishnamurthy V. (1966). On the number of topologies
Krishnamurthy
V. (1966).
the number
of topologies
Conclusions
on
a finite set. Amer.
Math. On
Montly
73, 154-157.
ve algo- on a finite set. Amer. Math. Montly 73, 154-157.
algo- Sarria, H., Roa L. & Varela R. (2014). Conexiones
evehave:
We have seenRoa
how L.
the&
considered
matrices
make
easier the
Sarria,
Varela
R.
(2014).
Conexiones
e have:
entre los H.,
espacios topológicos
finitos,
las
fd relaciones
y las
topological
studytopológicos
in finite spaces
when
we
use submodular
entre
los
espacios
finitos,
las
fd
relaciones
y las
funciones submodulares. Boletı́n de Matemáticas, 21(1),
functions,
achieving
to
eliminate
the
need
to
draw
Hasse
funciones
submodulares.
Boletı́n
de
Matemáticas,
21(1),
31-50.
diagrams to find its values as was worked in (Abril, 2015)
31-50.
, 0)
and (Sarria
et al., On
2014).
future works,
it can
be
Shiraki,
M. (1968).
finiteIntopological
spaces.
Rep.
, 0)
Shiraki,
M.
(1968).ofUniv.,
On
finite
topological
spaces.
Rep.
studied
the shown
algorithms and
try to
find
Fac.
Sci.complexity
Kagoshima
1, 1-8.
Fac.
Kagoshima
Univ., 1,
1-8. could be characterized
otherSci.
topological
concepts
which
Stong R. E. (1966). Finite topological spaces. Trans.
from this
pointFinite
of view.
Stong
R. matrix
E. Soc.,
(1966).
topological spaces. Trans.
Amer.
Math.
123(2), 325–340.
Amer. Math. Soc., 123(2), 325–340.
), 0)
), 0)
d for any
d
for any
values
in
values
in
he above
he above
asier the
asier
the
modular
modular
aw Hasse
aw
il, Hasse
2015)
2015)
til,can
be
ty can
be
to find
y
to
find
acterized
acterized
136
Abril J. A. (2015). Una aproximación a la noción de
homotopı́a entre espacios topológicos finitos desde las funciones submodulares (Trabajo Final de Maestrı́a). UniverRev. Acad. Colomb. Cienc. Ex. Fis. Nat. 41(158):127-136, enero-marzo de 2017
sidad Nacional de Colombia.
Facultad de Ciencias. Bodoi: http://dx.doi.org/10.18257/raccefyn.409
gotá, Colombia.
Alexandroff P. (1937).
(N.S.)2, 501-518.
Diskrete Räume.
Mat.
Sb.
Cuevas Rozo J. L. (2016). Funciones submodulares y
matrices en el estudio de los espacios topológicos finitos
(Trabajo Final de Maestrı́a). Universidad Nacional de
Colombia. Facultad de Ciencias. Bogotá, Colombia.
Krishnamurthy V. (1966). On the number of topologies
on a finite set. Amer. Math. Montly 73, 154-157.
Sarria, H., Roa L. & Varela R. (2014). Conexiones
entre los espacios topológicos finitos, las fd relaciones y las
funciones submodulares. Boletı́n de Matemáticas, 21(1),
31-50.
Shiraki, M. (1968). On finite topological spaces. Rep.
Fac. Sci. Kagoshima Univ., 1, 1-8.
Stong R. E. (1966). Finite topological spaces. Trans.
Amer. Math. Soc., 123(2), 325–340.

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Computation of matrices and submodular functions values

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