1. This research was partially supported by the Air Force Office of
Scientific Research under Contract No. AFOSR-68-1415, University of North
Carolina at Chapel Hill, Department of Statistics.
2. A portion of this work was undertaken while the authors were
attending the National Science Foundation Advanced Science Seminar on Combinatorial Theory at Bowdoin College, Brunswick, Me., during the summer of 1971.
3. and Department of Mathematics, University of North Carolina at
Chapel Hill.
Et.e£NTARY
AND
STROOG MAPs
T~SVERSAL C£a.t:ml ES l , 2
T. A. Dowling and D. G. Kelly3
Department of Statistics
University of North CaroUna at Chape 1. Hi 1.1.
Institute of Statistics M1meo Series No. 821
April, 1972
I
ELEMerrARY STRONG r1L\ps AND TRANSVERSAL GECH:TRIES ,2
T. A. Dowling and D. G. Kelly
f\BSTRACf
Let
same set
H ~ G be a strong map between two combinatorial geometries on the
X.
The rank function, flats, and independent sets of
characterized in terms of a factorization of
maps.
~
When
sentation of
H is the free geometry on
X,
G are
H ~ G into elementary strong
these results lead to a repre-
G as the basis intersection of a family of transversal geom-
etries, and dually, as the basis intersection of a family of principal
geometries.
E1..EMENTARY STRtlJG
MAPs
AND TRANSVERSAL GEQM£TRIES
T. A. Dowling and D. G. Kelly
One of the more familiar classes of combinatorial geometries [6] is the
class of transversal geometries of Edmonds and Fulkerson [7].
Dually related
to these are the principal geometries of Brown [1,2,3] (called 'F-products' by
him), which are defined in Section 1 below.
In this paper we obtain Brown's theorem on the orthogonality of transversal and principal geometries, as a special case of more general results
(Sections 2 and 3) on the factorization of the closure map of a pregeometry
into elementary maps.
These results also lead to theorems on 'representation'
of geometries in the following sense:
If, for a geometry
on
each
X such that a set
Gi ,
G on a set
X,
B is a basis of
then we say that
there is a family {G } of geometries
i
G if and only if
B is a basis of
G is the basis intepsection of the family
('Basis-family intersection' would be more precise.)
{G }.
i
Similar definitions can
be made for spanning-set intepsection, independent-set intepsection" etc.
Section 4 we show that an arbitrary pregeometry on a finite set
In
X is the in-
dependent-set intersection (and thus also the basis intersection) of a finite
family of principal pregeometries of the same rank on
X.
Dually, in Section
5 we show that an arbitrary pregeometry is the spanning set intersection (and
again also the basis intersection) of transversal pregeometries on the same set.
2
1.
~
Introduction.
In this section we collect various definitions and re-
suIts concerning pregeOmetries and transversals of families of sets.
For
further details the reader is referred to [6 t 9].
Let
L be a finite lattice.
but no element
z
exists with
An element
x < z < y.
A ahain in
covering the zero element.
C is an
Yt
An atom of
x-y
L.
If
chain.
Every saturated
Let
x-y
Xt
metric lattice.
A of
Xt
a,b
x-y
y. x v p
Xt
A £ B)
and
L is geop $ x.
Xt
X is a set of subsets
G are closed sets or fiats.
X is the minimal flat containing
A £ X,
x
L is satu-
chain in
for some atom
G on
A,
The cZosure of any
denoted
X is under consideration).
8 to G is a closure operator on 8 (i.e.,
€
L which is
and which, when ordered by inclusion, is a geo-
Members of
A £ B implies
C of
A finite lattice
A pregeometry (matroid)
more than one pregeometry on
from
x < y
L is an element
8 be the lattice of subsets of
X be a finite set and let
which contains
subset
if and only if
if
chain in a geometric lattice has the same cardinality.
ordered by inclusion.
of
x
x
C has minimal element
An
rated if it preserves the cover relation of L.
metria when y covers
L aovers
of
L is a subset
linearly ordered by the order relation of
and maximal element
y
A
(or
The map
A £ A,
I = A,
AG
A ~
A
and
which satisfies in addition the exchange property:
a
€
Aub
that the flats covering any flat
but
a
~
A,
then
b
€
Aua.
if
A consequence is
A partition the elements of
geometry on X if the empty set and all singleton subsets of
if
X-A.
G is a
X are closed.
8 is the free geometry on X.
The rank
r(A)
one t of all saturated
(or
~-A
rG(A»
of any subset
chains in
G.
A of
The rank of
X is the size, less
G is the rank of
X.
Flats of rank one, two, three are points, lines, planes, respectively, of
G.
3
The nuZZity of a subset
cardinality function.
A subset
A of
X is
The nullity of
cardinality, equal to the rank of
A spans
B.
An isthmus of
A 2 B,
B if
A spanning set of
B.
A basis of
B.
X.
X:
G on
A loop of
metry
G on
X iff
is the
X.
n(A)
B,
= O.
All
have the same
X.
A
A contains a basis of
An element contained in
B is isthmus-fpee.
G is an element of
~,
G.
X is characterized by any of the following families
flats, independent sets, bases, spanning sets.
1 of subsets of
set
X.
1·1
G is a basis of
If none exists,
i.e., an element contained in no basis of
of subsets of
called bases of
B,
G is a spanning set of
G is an isthmus of
A pregeometry
where
G is the nullity of
or equivalently, if
B is an isthmus of
every basis of
IAI - r(A),
X is independent (or G-independent) if
A of
maximal independent subsets of any set
subset
=
n(A)
A nonempty
X is the set of independent sets of a (unique) pregeoI
B and all maximal I-sets con-
is an order ideal in
tained in any set are of the same cardinality.
Corresponding to a pregeometry
G*
on
X.
The bases of
G*
X is its opthogonaZ ppegeometry
are the set complements of the bases of
Clearly
(G*)*. G and the rank of
between
G*
and
G on
G*
is the nullity of
G.
G.
Other relations
G include
(a)
A spans G* iff X-A is G-independent,
(b)
A is a G*-flat iff X-A is isthmus-free in G,
(c)
a is an isthmus of G* iff a is a loop of G,
(d)
r*(A)· n(X) - n(X-A) • IAI + r(X-A) - r(X),
(1.1)
where
r*
Let
4It
is the rank function of
G*.
H, G be two pregeometries on
X.
is an H-f1at, then the identity function on
If
H 2 G,
i.e., if every G-flat
X extends to a stpong map [6,8]
4
from
H to
is that
G and
-G 2 A
-H
A
G is called a quotient of
for every subset
A of
X.
H.
An
equivalent condition
We denote this strong map, which
takes each H-flat to the minimal G-flat containing it (i.e., to its G-closure),
by
H ~ G.
If
only if
Since
G,
In particular, the canonical closure map of
H ~ G is a strong map, then
G· H.
The difference
rB(X)" lxi,
n
A strong map
n
elementary maps
H ~ G.
is the nullity of
G.
H ~ G.
B ~ G is the nullity of
H ~ G of nullity zero is trivial
.
IS
Gi l
aonstruation of Higgs [8].
of
~
Every strong map
H~ G
may be written as a composite
H
of
8
with equality if and
and one of nullity one is elementary.
G .. H),
of nullity
rH(X) - rG(X)
rH(x)
the nullity of the closure map
as defined earlier.
(since
~
rG(x)
G is
~
G
i
(i· l, .•• ,n),
G
using for example the lift
Such a sequence we call an elementary faatoriaation
It is not unique in general.
Elementary maps may be factored further in the category of all pregeometries and strong maps.
that an elementary map
element extension [5,6]
added element
e.
A theorem of Brylawski [4] and Higgs [8] provides
H ~ G factors as an injeation of
K of
H into a single-
followed by a aontraation [6] of the
H,
There is consequently a one-one correspondence between
single-element extensions of
H and quotients of
H under elementary maps.
We describe below the basic facts we need concerning single-element extensions
[ 5] and the corresponding elementary maps.
A modular aut in a pregeometry
in
H with the property that if
fimum
A A B,
then
A A B is in
be the set of H-flats not in
M,
H on
X is a nonempty order filter
A, B are in
M.
M
M and each covers their in-
Given a modular cut
but covered by flats in
M in
M.
H,
let
eM
Then every flat
5
~
of
CM is covered by a unique member of
on
Xu e
M.
A single-element extension
K
H is determined by a modular cut M in H and conversely.
of
The corresponding elementary quotient
G· K/e
of
H on
X is then given by
G • H-CM• The elementary map H ~ G fixes all flats of H-C , and takes
M
each flat of C to the unique M-flat covering it. One exception to the
M
foregoing must be noted. If M = H, then C is empty and the map H ~ G is
M
trivial, not elementary.
that of
The rank function of
G
(when
M ~ H)
is related to
by
H
AH
if
is in
M.
(1. 2)
if not.
We use the following notation to denote a trivial or elementary map.
E is any antichain of subsets of
one or more members of
H
M,
! G.
X such that the set of H-flats containing
E is a modular cut
mentary (or trivial) quotient of
If
M of
H,
and if
G is the ele-
H under the corresponding map, we write
We could of course always take
E to be the set of miminal flats of
but it will be convenient for our purposes not to assume the members of
are flats of
member of
H.
E
H {G is trivial if and only if -H
¢>
contains a
Note that
E.
M defining an elementary (or trivial) map
If the modular cut
a principal filter of
G,
the map
H ~ G will be called a principat map.
Such a map is specified by any subset
and may be written
E G.
H~
principal maps.
E spanning the generator of M in H,
We call a pregeometry
geometry if the closure map B
~
H ~ G is
G on
X
a principal, pre-
G admits an elementary factorization into
Such pregeometries were first investigated by Brown [1,2,3],
who called them IIF-products ll •
6
Let !
~
= (E1 ,E 2 , .•• ,Em)
be a finite family of subsets of
A transversaZ of
need not be empty, nor need they be distinct.)
m-subset
{x l ,x ' ••. ,xm} of
2
partiaZ transversaZ of !
X for which
xi
E
i
£
for
[9] of the "marriage" theorem of P. Hall provides that a subset
k S m and choice of
!
of size
i , •.• ,i with
1
k
IA
k
j-1
i
E
i
E is an
E.
A
A corollary
A of
X con-
n S m if and only if for every
1 S i
I
U E
n
(The
i · 1, ••• ,m.
is a transversal of a subfamily of
tains a partial transversal of
X.
~
1
< ••• < i
k
S m,
the inequality
k - (~n)
j
holds.
It was observed by Edmonds and Fulkerson [7] that the set of partial
transversals of a family !
satisfy the conditions given above for the inde-
pendent sets of a pregeometry on
pregeometry~
of
T(!)
2.
Such a pre geometry is called a transversaZ
X.
and will be denoted here by
T(E),
or
are the maximal partial transversals of
T(E , ••• ,Em).
1
The bases
~.
Rank and closure for elementary factorizations.
We consider through-
out this section a fixed elementary factorization
E
H ..
(2.1)
of a strong map
•••
H + G of nullity
By (1.2) we have, for any
•
(2.2.)
(Observe that if
H-
n
~
G
n
..
G
n.
A£ X
_G I
I{i: 1 SiS n and A i 1 contains a member of Ei }
B, then r H - r G is the nullity function of G.)
A £ X,
Define, for
(2.3)
f(A)
•
I{i: 1 s i s n and A contains a member of fi}l.
The following proposition is then immediate.
Proposition 2.1: Let A be any subset of X. Then
(1)
(ii)
0 s f(A) S rH(A) - rG(A).
If
A is a G-flat, then
Theorem 2.2:
For any subset
f(A). rH(A) - rG(A).
A of
X,
min
[rH(B) - feB)],
B 2 A
where
A is the G-closure of
Proof:
~
A.
rG(A). rG(A) • rH(A) - f(A)
B 2 A is arbitrary, then
by Proposition 2.1 (ii).
rG(A) S rG(B) S rH(B) - feB)
And if
by Proposition 2.1 (i).
o
Corollary 2.2a:
(2.4)
for each
Proof:
If
rG(A)
by
B
= A.
~
If
rG(A). rH(A)
feB) s rH(B) - rH(A).
B 2 A,
rG{A). rH(A)
rH(A),
then
if and only if
and
rH(A)
>
B 2 A,
then
rH(A)
rG(A) • rH(A) - f(A),
= rG(A)
S rH(B) - f(B).
and (2.4) is violated
0
Corollary 2.2b: A is G-independent if and only if A is H-1ndependent
and
for each
B 2 A,
f(B) S rH(B) - IAI.
Corollary 2.2c: If H· B in (2.1), then for any subset A of X
8
=
rG(A)
•
IAI - f(A)
min
lIBI - feB)].
B 2 A
and
A is G-independent if and only if
for each
Corollary 2.2d:
(Fl, ••• ,F )
n
If
feB) ~ IB-AI.
B 2 A.
is any permutation of
(El, .•• tEn)
for which the sequence
F
(2.S)
H •
o --1->
G'
G'
1
F
--.1...>
F
--!L> G'
n
is well-defined (in the sense that for
members of
Fi
Proof:
i · l, •••• n
the
form a modular cut which is not all of
Gi_l-flats
GI-l)'
then
G'n
is determined uniquely by
H and the set function
Neither of these depends on the order of the sequence
(El, ••• ,E ).
n
The hypothesis of Corollary 2.2d is of course quite restrictive.
the special case treated in later sections, in which each
the filter of flats above members of
E
i
Theorem 2.3: A is a flat of G if and only if
for each
•
If
f(A)
feB)
B
~
A,
feB) - f(A)
A is a G-flat and
=
B
<
~
E
i
f.
o
But in
is a singleton,
will be a principal filter and hence
a modular cut.
Proof:
G'n • G.
Theorem 2.2 provides that as long as (2.S) is a sequence of ele-
mentary maps,
(2.6)
above
rH(B) - rH(A).
A,
then
by Proposition 2.3 (ii),
by Proposition 2.3 (i),
9
and
Hence
f(B) - f(A)
If
<
A is not a C-flat, then since
=
rC(A). rC(A)
rH(A) - rC(A) - [rH(A)-rc(A)]
f(A) - f(A)
so that (2.6) is violated by
Corollary 2.3a:
If
H·
for each
When
H·
B,
C.
by Proposition 2.1,
0
B· A.
B in (2.1), then A is a C-flat if and only if
B ~ A,
the function
the nullity function of
we have
f(B) - f(A)
rH - r
C
<
IB-AI.
defined by (2.2) is, as noted above,
As such it is (lower) semimoduZar:
n(AuB) + n(AnB)
n(A) + n(B).
~
Although many of the results of this section for the function
f
defined by
(2.3) are similar to the corresponding results in terms of the nullity function
n,
is not in general a semimodular function.
f
3.
Elementary maps and partial transversals. Throughout this section,
we consider an arbitrary pregeometry
factorization of the closure map
C on
B ~ C.
X of nullity
n,
and a fixed
The proof of Theorem 3.3 will require
consideration of the more general case, in which trivial maps as well as
10
elementary maps are admitted in the factorization.
~
somewhat by the following observation:
if
G also.
For
E
then
E defines the
E defines the trivial map on
H if and only
-H contains a member E of
4>
at once.
H ~ G is any strong map, and
B defining the trivial map on H,
is an antichain in
trivial map on
If
The notation is simplified
E;
-G
since
4>
2
-H
the conclusion follows
4> ,
Accordingly, any trivial map appearing in a factorization commutes
with any elementary map immediately following it; the defining antichains of
the two maps may be left unchanged.
(3.1)
B
Thus let
E
••• --!L..> G
•
n
B
be a fixed factorization of the closure map
m- n
trivial maps, where
of generality assume that
m ~ n.
El, ••• ,E
n
apply the results of Section 2 to
G with
We denote by
product set
•
!.
II
m
~
G into
n
•
G
elementary and
define elementary maps, and
Thus
f(A)
E
--!L> G
i>
By the above remarks, we may without loss
En+l, ••• ,Em define trivial maps.
(3.2)
En+1
Gn • G
n+l · ••• • Gm • G,
f
and we may
defined by
I{i: 1 SiS n and A contains a member of EiJI.
(El, ••• ,E )
m
x E
2 x ••• x
a vector of subsets of
X,
and by
E the
Em.
Theorem 3.1: Suppose G is a pregeometry of nullity n on X and
(3.1) is a factorization of the closure map
B ~ G into n elementary maps
and
X is independent in
m- n
trivial maps.
Then a subset of
only if its complement contains a partial transversal of size
in
n
G if and
of every !
E.
Proof: We can assume that En+l, ••• ,Em define the trivial map, so that
~
(3.2) holds.
all
B
~
By Corollary 2.3c,
X - A,
X - A is G-independent if and only if for
11
(3.3)
feB)
IAnB I
s
As noted in the introduction,
A contains a partial transversal of size
n
E if and only if
of
k
(3.4)
k - (m-n)
IA n U Ei
S
j-l
We show (3.3) holds for
~
B
Suppose (3.3) holds for
E.
I
for all
k S m and
<
j
X -.A if and only if (3.4) holds for all
B
~
~,k,il, ..•
X - A and let
E in
,ik be given.
Define
k
B At most
m- n
of the
ij
(X-A)
exceed
U E. •
u
j-l
1j
so by (3.2),
n,
feB)
~
k - (m-n).
Then
from (3.3),
k - (m-n)
S
feB)
Conversely, suppose (3.4) holds for all
(3.3) for B closed in G,
~
k
=
IAnBI
S
IA n
U Ei ,.
jo:l
f.
in
j
It is sufficient to prove
for if (3.3) fails for
B but holds for
H,
then
feB) - feB)
IH-BI,
<
which contradicts Corollary 2.2 (c).
Denote
feB)
by
i.
Then there exist
1 S i l < ••• < i S nand
l
Ei
in
j
E
i
En+l, .•• ,Em define the trivial map on
of
e
~,
and hence of
B,
with
E
i
in
for
:::l
U E
i
j
j-l
j
0:
j
G,
there exist
E
i
for
.e.
B
Ei , ..• ,E i
of B with
1
l
l, •.. ,l. Further, since
subsets
u
subsets
i=n+l, ••• ,m.
m
U E ,
i
i=n+1
En+1 ,···,Em
Thus
12
so by (3.4)
IAnBI
A n [
~
l ""f(B).
j-l
o
Theorem 3.2: Let (3.1) be any factorization of the closure map B + G
into
n
elementary maps and
m- n
trivial maps.
Then the nullity
n
of
G satisfies
n
•
max{j: every! in
£
has a partial transversal of size j}.
Proof: Certainly every E in E has a partial transversal of size n;
for Theorem 3.1 provides that the complement of any G-independent set contains one.
So we need to show that there are
versals of size exceeding
Again assume that
~
contains some
Ei £
i.
E
i
in
Ei '
j
j
So no
E in
Iii
Denoting
by
r(~)
...
E , ••• ,E
i
i
1
k
j =l, ... ,k. But then
... Iii
E containing
transversal of size exceeding
~e
with no partial trans-
En+l, •.• ,Em define trivial maps in (3.1), so each
subsets
k
£
in
n.
o ...
Thus there exist
!
~
Ij~l
k,
we have by Corollary 2.2c that
k -
f(i).
i
with
of
m
E
i
U
j
U
i-n+l Ei
Ei1, •.• ,Eik,En+l, ••• ,Em
n.
1
s i
1
<... < i
k
S
n
and
I·
can have a partial
0
preceding two theorems provide a short proof of the following theorem,
due to T. Brown [3).
13
Theorem 3.3: Let -E· (El, ••• ,E)
be an arbitrary vector of subsets of
m
~
X,
T(!)
its induced transversal pregeometry, and
PC!)
the principal pre-
geometry defined by the sequence
E
l
o ->
8 ..
(3.5)
Then
T(!)
G
and
Proof:
P<,~,>
E
P
1
--L>
E
m
Pm •
--:>
P(E).
are orthogonal pregeometries.
Observe first that since the
E
i
are not required to be closed
and trivial maps are admitted, the factorization (3.5) is well-defined.
PC!)
have nullity
n.
Then by Theorem 3.2,
follows from Theorem 3.1 that
A spans
n
is the rank of
X - A is independent in
PC!)
Let
T(!).
It
if and only if
0
T(!).
Corollary 3.3a:
of ! . (El, .•. ,Em),
If ! .. (Fl, ••• ,F )
m
then
is any permutation of the members
P(!) .. PC!).
Two other easy consequences of Theorem 3.3 are the following results,
well-known in transversal theory.
Corollary 3.3b:
rank
then there exist
n,
Proof:
the
m- n
I
H· T(El, ••• ,E )
m
i l < ••• < in
is a transversal pregeometry of
such that
H" T(E
In the factorization (3.5) of the closure map
E
i
'8
defining trivial maps.
Corollary 3.3c:
and
If
If
X - Ei ,
8
8
~
H* .. P(El, .•• ,Em)
~
H*,
i
).
n
delete
then
is a transversal pregeometry,
H" T(El, .• ·,EiUI, ••• ,Em).
Proof: We may assume i" m by relabeling if necessary.
closure map
, ••. ,E
1
o
H· T(El, .•. ,Ei, ••. ,E )
m
is any set of isthmuses of
i
may be factored
Then the
14
B
where
F
...
E
E
o ---L>
G
is any set betweeri
P
1
2
->
P
m-l
...L> Pm = H*,
Em and its Pm- l-closure.
But the Pm-l-closure
of
E is the same as its H*-closure, and the H*-closure of
m
of
E with all isthmuses of
m
X - E in
m
E is the union
m
0
H.
4. Basis intersections of principal pregeometries. In this section, we
again omit trivial maps and consider an elementary factorization
B
(4.1)
...
E
E
G _l_~ G _2_~ ... -En
> G
o
1
n
of the closure map of a pregeometry
Section 3, therefore, apply with
G of nullity
m'" n.
E has a transversal, so every
E in
nullity
=
G
n
on
X.
The results of
In particular, by Theorem 3.2, every
T(!)
has rank
n
and every
n.
The next theorem states that every pregeometry on
set intersection of principal pregeometries on
X is the independent-
X.
Theorem 4.1: If G is a pregeometry of nullity n on X,
a family
PC!)
P
of principal pregeometries of nullity
n
on
X,
there exists
such that a sub-
set is G-independent if and only if it is P-independent for every
P
in
P.
Proof: Let (4.1) be an elementary factorization of B + G. By Theorem
3.1, a subset
spans
T(E).
X - A is G-independent if and only if, for every
But the orthogonal of
by Theorem 3.3, so
in every
PC!).
T(E)
E in
is the principal pregeometry
£'
P(!) ,
X - A is G-independent if and only if it is independent
Thus
p ... {peE): E
€
E}
is the desired family.
0
A
15
peE), or simply P,
In what follows we use
~
{Pc!): E
to denote the family
of principal pregeometries.
€ E}
Corollary 4.la: A subset of X is a basis of G if and only if it is a
basis of every
P
in
P.
The rank function of
G is determined by the rank functions of
P by
means of
Theorem 4.2: The rank in G of any subset of X is its minimum rank
over all
P
in
P.
Proof: Let r, r* denote the rank functions of G, G* respectively,
and for any!
T(!) ,
I,
in
respectively.
let
r E , r E*
For a subset
denote the rank functions of
A of
X,
P(!) ,
by (l.ld),
•
so
•
(4.2)
IAI - n + min rE*(X-A).
E
For the transversal pregeometry
(4.3)
where
(4.4)
= Ix-AI
rE*(X-A)
0E
-
-
T(!) ,
max
B
::>
A
the rank function is given [9, 10] by
0 (X-B),
E
is the deficiency function [10 ]
=
IX-BI - I{i: (X-B) n Ei ; $}I
•
IX-BI - n + I{i: B 2 E.}I·
~
Thus by (4.2), (4.3) and (4.4),
16
= Ixi -
min rE(A)
E
-
max max (IX-BI + l{i:B2Ei}l)
E B2A
Ixi - max (IX-BI + max l{i:B2Ei }I) .
•
B2 A
Now for fixed
E
B,
= feB),
max I{i: B 2 Ei}1
E
so
min rE(A)
E
Ixi - max (IX-BI +
•
-
f(~»
BaA
•
(IBI - feB»~
min
B 2 A
•
by Corollary 2.2c.
~
rCA),
0
Theorem 4.3: Every flat of G is a flat of the same rank in some P in
P.
Proof: Let A be a flat of G. Then n(A)· f(A),
(4.5)
IAI - f(A)
for all
B
~
A.
and by Theorem 2.3,
IBI - feB)
<
For any
I,
E in
define
f E by (2.2) for the principal
factorization
E
8
Clearly
f(A).
fE(B)
Then if
S
G _1_~ P
o
•
feB)
B
~
A,
1
E
E
2
--~
for any subset
-!L> P
n
B.
Choose
E in
we have by (4.5),
IAI - fE(A)
•
=
IAI - f(A)
<
IBI - feB)
S
IBI - fE(B),
P(]).
E so that
fE(A)
=
17
so
A is a flat of
A in
P(~).
5.
Since
P(~),
by Corollary 2.3a.
fE(A)
= f(A),
Thus,
is the nullity of
= r(A).
rE(A)
Basis intersections of transversal pregeometries.
section are dual to those of Section 4.
section.
All notation will be as in that
Our next theorem states that every pregeometry on
spanning-set intersection of transversal pregeometries on
Theorem 5.1:
spans
If
H is a pregeometry of rank
n
on
T of transversal pregeometries of rank n on X,
family
H if and only if it spans every
Proof:
The orthogonal pregeometry
T
in
G
= H*
X spans
H if and only if
3.1 is equivalent to
of
=
{T(~):
E
€
E}
in
£'
is the desired family.
G,
so each
T(~)
Corollary 5.la:
basis of every
T
in
A subset of
there exists a
such that a subset
n,
so
A of
which by Theorem
E.
E in
has rank
n,
Thus
and
X is a basis of
{T(~):
E
€
E}.
H if and only if it is a
T.
The next two theorems are the analogues of Theorems 4.2 and 4.3.
Theorem 5.2:
over every
T
in
The rank in
T.
X
0
P, we let T(I) , or simply T, denote
As with
X,
Then a subset
A containing a transversal of every
~
X.
H has nullity
X - A is independent in
contains a transversal of every
X is the
T.
its closure map has a factorization of the form (4.1).
T
The results in this
H of any subset of
X is its minimum rank
18
Proof:
For any subset
Theorem 5.3:
X - A of
Every flat of
X,
we have by (4.2) that
=
n - IAI + min rE(A)
E -
=
n - IAI + r(A)
=
IX-AI + r(A) - r(X)
=
r*(X-A).
0
H is a flat of the same rank in some
T in
T.
Proof:
By (l.lb),
isthmus-free set of
(5.1)
for every
a
€
=
r(A-a)
A,
a
in
B 2 A-a,
X - A is a flat of
G = H*,
H if and only if
A is an
that is, if and only if
r(A)
A.
By Corollary 2.2c, (5.1) holds if and only if for every
there exists
Al 2 A such that
(5.2)
(In this case, the G-closure of
flat of
T(K)
A serves as
if and only if for all
a
€
A,
AI')
Similarly,
B 2 A-a,
X - A is a
there exists
A 2 A
2
such that
Suppose
f(A)
since
X - A is a flat of
= n(A),
fE(B)
H,
and we can choose
s;
(5.2), for any
f(B)
a
€
and let
E in
for any subset
A,
B 2 A-a,
B,
A = A,
2
E such that
the G-closure of
fE(A)
=
f(A) .
A.
Then
Then
we have, by the remark following
19
Inl -
fE(B)
~
IBI - f(B)
IAI
= IAI
~
Hence
X - A is a flat of
T(~).
- f(A)
- fE(A).
Now
r*(X-A)
=
r(A)
=
r(A)
=
r E*(X-A)
i f and only i f
By Theorem 4.2,
but
r(A)
= IAI - f(A)
= IAI - fE(A)
~
=
min (IB I
B .2 A
-
fE(B»
o
Observe that the statements of Corollary 5.la, Theorem 5.2, and Theorem
5.3 are just those of the corresponding results of Section 4, with "principal"
and "nullity" replaced by "transversal" and "rank", respectively.
analogy does not hold for Theorem 5.1 with respect to Theorem 4.1.
The same
That is,
it is not necessarily true that every pregeometry is the spanning-set intersection of principal pregeometries or the independent-set intersection of
~
transversal pregeometries.
For example, in Theorem 5.1, an H-independent set
is T-independent for every
T in
T,
but the converse is not necessarily true.
20
REFERENCES
[1]
Terrence J. Brown, Deriving Closure Relations with the Exchange
Property, I and II. Preprint, University of Missouri at Kansas
City 64110.
[2]
Terrence J. Brown, Finitary Exchange Closures and F-Products.
University of Missouri at Kansas City 64110.
[3]
Terrence J. Brown, Transversal Theory and F-Products.
University of Missouri at Kansas City 64110.
[4]
Thomas H. Bry1awski, The Tutte-Grotnendieck Ring.
Dartmouth College, Hanover, N.H., 1970.
[5]
Henry H. Crapo, Single-Element Extensions of Matroids.
J. Res. Nat. Bur. StandardS Sect. B 69B (1965) 55-65.
[6]
Henry H. Crapo and Gian-Carlo Rota,
Preprint,
Preprint,
Ph.D. dissertation,
On the Foundations of Combinatorial
Theory: Combinatorial Geometries (preliminary edition), M.I.T.
Press, Cambridge, Mass., 1970.
[7]
Jack Edmonds and D.R. Fulkerson,
Transversals and Matroid Partition.
J. Res. Nat. Bur. StandardS Sect. B 69B (1965) 147-153.
J. Comb. Thy. 5 (1968)
[8]
D.A. Higgs, Strong Maps of Geometries.
185-191.
[9]
L. Mirsky, Transversal Theo~. Academic Press, New York, 1971
(Vol. 75 in Mathematias in Saienae and Engineering.)
[lOJ
O. Ore, Graphs and Matching Theorems.
625-639.
Duke Math. J. 22 (1955)
FOOTNOTES
AMS Subject Classifications:
Primary
0504, 0535
Key Words and Phrases: Combinatorial geometry, matroid, transversal, strong
map, elementary strong map.
1. This research was partially supported by the Air Force Office of
Scientific Research under Contract No. AFOSR-68-1415, University of North
Carolina at Chapel Hill, Department of Statistics.
2. A portion of this work was undertaken while the authors were
attending the National Science Foundation Advanced Science Seminar on Combinatorial Theory at Bowdoin College, Brunswick, Me., during the summer of 1971.