Research was partially supported by the U. S. Air Force under Grant No.
AFOSR-68-l406 and the National Science Foundation under Grant No. GU-2059.
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COM BIN A TOR I A L
MAT HEM A TIC S
YEA R
If
5
9
8
7
Febr~
1969 - June 1970
\!l.~TaHNG THEORY FOR Ca-1BlNATORIAL GEOMETRI ES
by
i'llarti n Aigner
and
THOMAS A. DOWLING
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 600.28
~fay
1970
MATCHING THEORY FOR COMBINATORIAL GEOMETRIES 1
by
Martin Aigner
Thomas A. Dowling
Department of Statistics
University of North Carolina at Chapel Hill
1. I NTROIXJCTt ON ,
A geometric reZation we define as a triple
G(S), G'(T)
and
(G(S), R, G' (T»,
are pregeometries (matroids) on point sets
R S SxT
is an arbitrary binary relation from
S
S, T, respectively,
to
T.
example of a geometric relation is a bipartite graph, in which
are free geometries.
where
The simplest
G(S), G'(T)
In the present paper, we consider several questions
which originated historically with finite bipartite graphs, or with their
equivalent representation as a family of subsets of a finite set.
Some
classical results of matching theory for bipartite graphs are extended to
geometric relations.
A matching in a geometric relation
of
R,
set of
G' (T)
(G(S), R, G'(T»
the elements of which define a bijection
G(S)
to an independent set of
G' (T).
<PM
is a subset
M
from an independent
We assume that both
G(S),
have finite rank, from which it follows that any matching is finite.
A maximum matching is one of maximum cardinality.
Bya support of
closed sets in
(G(S), R, G'(T»,
G(S), G' (T),
we understand a pair
respectively, which cover
R
(e, D)
of
in the sense
Research tJas partiaZly supported by the U. S. Air Foree under Grant No.
AFOSR-68-1406 and the NationaZ Science Foundation under Grant No. GU-2059.
1
Some of the results contained in this paper appeared previously as a
research announcement [1].
2
that for all
(c,d) € R,
either
c €
e or d
are called the e'tements of the support
(C, D).
€
D holds.
The rank
The flats
p(e, D)
C, D
is the
sum of the ranks of its elements, and a minimum suppozrt is one of minimum
rank.
For bipartite graphs our definitions reduce to the usual ones.
In
this case, the concepts of maximum matching and m1nimtml support are related
by the well-known Konig-Egervary Theorem.
metric relations (Theorem 2) in §3.
We extend this theorem to geo-
The proof rests on a characterization
(Theorem 1) of a maximum matching in terms of the non-existence of an
"augmenting chain".
(cf.
The latter concept originated with bipartite graphs
Berge [2]), where it is associated with the ''Hungarian method" for
finding a maximum matching.
Our definition extends the notion of an aug-
menting chain to geometric relations by means of the MacLane-Steinitz exchange property.
In §4, Ore's [8] definition of a dEficiency function on subsets of
S,
for the case of a bipartite graph, is generalized to a geometric relation.
The sets of maximal deficiency, called critical sets, are shown to form a
ring.
The open sets in this ring form a distributive sublattice of the
lattice of open sets (Theorem 3).
The notion of maximal deficiency is applied in §5 to obtain an expression for the cardinality of a maximum matching (Theorem 4), a result
proved by Ore [8] for bipartite graphs.
As a corollary to Theorem 4, we
obtain a generalization of the Marriage Theorem of P. Hall (see e.g. [6])
and Rado [9].
The minimal critical open set is characterized in terms of
the effect on the maximal deficiency when contracting
G(S)
In §6, we investigate the structure of minimum supports.
by a point.
The relation
R induces a dual Galois connection between the lattices of closed sets in
3
G(S), G I (T) ,
for which the elements of irredundant supports (defined in
§6) are the coclosed elements, with the canonical anti-isomorphism between
the quotient lattices specifying the corresponding elements in each such
support.
Among these, the elements of minimum support are shown to form
anti-isomorphic distributive sublattices of the lattices of closed sets of
G(S), G' (T) •
As a consequence, we show that the minimum supports exhibit
a distributive lattice structure (Theorem 5).
2.
PRELIMINARIES.
Our primary reference for notation, definitions, and terminology is
Crapo and Rota [4].
We smnmarize some basic concepts in the present section
which will be needed later.
A ppegeometry
consists of a set
G(S)
operator
A 1+ A on subsets of
(2.1)
Exchange ppoperty:
subset
(2.2)
A £ 5,
if
a
£
S
A set
is closed.
ASS
AO
III
enjoying the following properties:
For any elements
Aub,
Finite basis pPOpepty:
A such that
O
together with a closure
S
a
~
A,
a, b
then b
Any subset
A
=S
£
£
S,
and for any
Aua.
has a finite subset
A.
A ... A,
is cZosed if
A pregeometry
and open if its complement in
S
is open if the null set is closed, i.e. if
G(S)
S is open.
A combinatorial, geometry (briefly, a geometry) is an open pregeometry
G(S)
for which the elements
a
to:
S,
ically associated with any pregeometry
called points, are closed.
G(S)
is a geometry G(SO)
Canonwhose
4
points are equivalence classes of elements of
relation
a .... b
Given a pregeometry
minor G[A,B]
a'"
if and only if
G(S)
S
-~,
under the equivalence
b.
and subsets
A, B of
is a pregeometry on the difference set
S with
A So B the
B - A with closure
operator
(2.3)
J [A,B] (C)
.
(CuA n B) - A
for
C S B-A•
Of particular importance among minors are the reductions to sets
J[~,B] (C)
(2.4)
..
Cn
B
and the contractions to sets
(2.5)
J[A,S] (C)
The set
closure.
ASS
..
for
B S S,
C So B,
S - A,
CuA - A
for
C S S-A.
B S S is independent if it is a minimal set with given
By (2.2) any independent set is finite.
The rank
rCA)
of a set
is defined as the cardinality of the largest independent subset of
The rank function
r
satisfies the (upper) semi-moduZar inequality
(2.6)
For a minor
(2.7)
G[A,B]
r[A,B] (C)
where
•
of
G(S),
the rank function is
r(AuC) - rCA),
C S B-A.
It follows from (2.7) that if
then it is also independent in
C is independent in the minor GrA,B]'
G(S) •
The closed sets, or !tats, of a pregeometry G(S),
clusion, form a geometric lattice
L(S)
in which
ordered by in-
A.
5
(2.8)
The lattice
sets.
L(S)
is anti-isomorphic to the lattice
of open
A canonical anti-isomorphism is provided by complementation
C ... S-C
with respect to
S.
The cardinality of any finite set
3.
M(S)
A will be denoted byv(A).
Pu:lerrING OWNS.
Throughout this paper, we consider an arbitrary (but fixed) geometric
relation and denote it by
(G(S), R, G'(T».
G' (T)
r, r',
will be denoted by
~lation
(G(S), R, G'(T»
of
R' £ TxS
is defined by
G(S),
The converse geometric
respectively.
is the relation
(b,a)€R'
The rank functions of
(G'(T), R', G(S»
if and only if
(a,b)€R.
problems we consider will be symmetric with respect to
where
Most of the
G(S)
and
G'(T).
The distinction between a geometric relation and its converse in such cases
is unnecessary, but it will be convenient to distinguish the two on some
occasions.
The relation
subsets of
(3.1)
to subsets of
R(A)
The function
and
S
R defines a function, which we also denote by
•
R
B(T),
(3.2)
preserves unions,
{b€T:
T,
(a.b)€R
where for any subset
for some
R,
A £ S,
a€A}.
is order-preserving between the Boolean algebras
B(S)
from
6
(3.3)
but not necessarily intersections,
It follows from (3.2) to (3.4) and the semi-modular inequality (2.6)
for
r'
sets of
that the composite fmction
is upper semi-modular on sub-
r'R
S:
The analogous definition and properties hold for the converse relation
R' •
A matching M in
(G(S), R, G'(T»
~M: ~B
by its corresponding bijection
the independent sets
chain
of
Given a matching
with respect to
2n+1
(n~O)
~M(a»:
lfl : A-+B
M
~
i
~
n.
o
~
i
~
o
€
s-:A,
b~+l
€
T-B.
n.
Thus
(G(S), R, G'(T»,
distinct ordered pairs such that
(3.7)
a
in
U.
a€A}.
M is a sequence
1
(3.8)
when it is necessary to specify
A, B which are matched by
M = {(a,
Definition.
will be denoted alternatively
an augmenting
7
aI
i
E:
A,
i
1-1
a' t (A - U a ) u U a '
1
j=l j
jal j
bI
E:
a,
bi ~ (B -
,
(3.9)
i
Note that if both
ai
= ai'
bi
=b i
for
i
i-I
U b j ) u U bj'
jal
j-l
G(S), G'(T)
1
$
i
$
n,
1.
i
$
n.
so that our definition reduces to that
(G(S), R, G' (T»
We shall prove that
M is a
if and only if there does not exist
an augmenting chain with respect to M.
PRoPosITION
$
are free geometries, (3.9) 1mp11es that
of an augmenting chain in a bipartite graph.
maximum matching in
1
The first step in the proof is
If a matching M admits an augmenting chain, i t is not
maximum.
PRooF:
Let the chain be given by (3.6), and define
p'
o
=
$
i
:s
n} •
A straightforward inductive argument, using (3.9) and the exchange property,
shows that
A'
i
B'
i
=
(A -
i
U aj
j=l
=
(B -
i
i
u
U
b'
U b >
j=l j
jal j
>u
1
U a'
jal j
and
are independent sets with closures
Thus by (3.8), the sets
AI
•
AI ua'
n 0'
A, H,
respectively, for
B' .. B'ub I
n n+l
1 :s i :s n.
are independent sets,
8
each of cardinality
v (M)+l.
It follows that
M'
is a matching of cardinality
PROPOSITION
2.
If
(M-P) u P'
a
v (M)+l,
so
is a matching and
M
M is not maximmn.
(C,D)
is a support, then
v(M) S p(C,D).
PROOF:
By definition,
R(S-C) £ D.
v(M)
Thus
"'"
v (A)
"'"
v(AnC) + v(An(S-C»
a
v(AnC) +
=
r(AnC)
S
r(AnC) + r'(R(An(S-C»)
S
r(C) + r'(R(S-C»
S
r(C)
o
p(C,D).
+
~(~M(An(S-C»)
r'(~M(An(S-C»)
+ r' (D)
To prove the converse of Proposition 1, we require several lemmas
valid for any pregeometry.
l.Ef.t1A 1.
If
Bl' B
2
are subsets of an independent set
B,
then
The proof is straightfo:rward.
LEIYMA 2.
If
B
is an independent set, and
is the unique minimal subset of
D So
ii,
B whose closure contains
then the set
D.
9
PRooF:
bEB
Suppose
such that
l
B 1
of
Thus
D S B ,
2
B So B-b.
2
D S B ,
2
B So B.
2
where
D So B-b,
But then
B S. B
2
B
1
If
imply
B 2 B
2
1'
• ,n
(B-b')
!
B ,
2
there exists a point
contradicting the definition
To prove that
D S B ,
1
apply Lemma 1:
b 'EB-B
~
3.
Let
1
B be an independent set and
a strictly increasing sequence of subsets of
(1
~
i
~
n)
B.
Suppose
are points such that
(3.10)
(3.11)
Then
i
b'
i
for
1
PRooF:
~
i
Let
~
n.
~
(B -
Ub ) u
j=l j
i-I
U b'
ja 1
j
b , bI
i
we
10
We first show that
Bi· B •
i
By (3.11),
change property (2.1) imply the result for
induction.
bi (Bl-b ,
l
i 1lI'1.
so (3.10) and ex-
~e
proof proceeds· by
Let
C
i
..,
(B
-
i
i-I
i-I
U b ) u U b'
j"'l j
j-l j
Then
C
i
...
...
(Bi-B _ ) u Bi_l
i l
(Bi-B _ ) u Bi_l
i l
...
(Bi-B _ ) u B _
i l
i l
...
(Bi-B _ ) u B i l
i l
...
Bi
(by hypothesis)
'
and by a similar argument,
Ci -b i
-
It follows now from (3.11) and the exchange property that
...
...
Thns
-B'
i·
11
i
i-I
Ub ) u Ubt
j=l j
j=l j
(B -
=
and the lemma follows by (3.11).
~
4.
Let
A
be an independent set and
a strictly increasing sequence of subsets of
(1 s i s n)
A.
Suppose
are points such that
(3.12)
(3.13)
Then
i
~
at
U aj )
(A -
j=l
i
for
i-I
U a'
j=l j
1 SiS n.
PRooF:
up to
U
By (3.13),
i-I,
where
ai
d A-aI'
2 s i s n.
so assume inductively that the lemma holds
Then
i-I
C
1
is independent.
.,.
i
U aj )
j=-l
We can write
C
(A -
iii
and apply Lemma 1, obtaining
i-I
U
U a'
j-1
j
12
Since A-Ai_l-a S A-a ,
i
i
proof is complete.
PRCPOSITION
3.
it follows by (3.13) that
If a matching
cjlM: A+B
1 ( Ci-Si ,
and the
a
does not admit an augmenting chain,
(C ,D) , where
there exis ts a support
C S A,
PROOF:
Let
EO'" S-A.
Then
s
R(E )
O
a,
for otherwise there would exist a
trivial augmenting chain consisting of one element of R.
minimal subset of
Let
Al
E '" S-A-A • In general, having defined
l
l
as the minimal subset of B such that R(E _ ) n
i l
B
i
set
= cjlM-1 (B i ),
n
fi,
but
and so
Ei '" S-A-Ai •
cjlM(A - ) $. B-b
i l
Since
for any
A_ S E _ ,
i l
i l
B be the
1
R(EO) S Bl •
Ei _ l ,
as
B ,
i
we
and
cjlM(Ai - l ) £ R(E i _ l )
bE:B _ • Thus by Lemma 2,
i l
B_ S B ,
i 1
i
Ai - S Ai'
l
E _ S E • It is clear, moreover, that each of the sei l
i
Ai' B , E is strictly increasing up to and including some index
i
i
quences
m,
defined according to Lemma 2, such that
= cjlM-1 (B l ) ,
define
Ai
B,
Let
after which the process terminates.
Thus
but
for
0
~
iSm-I,
where
We shall prove that
w~se,
TL.:'l~
let
n
B '" 0.
O
R(E ) £
i
a
for all
be the smallest integer,
there e:r.:ists
(a~,b~l)
i,
1 S n S mt
E: R such that
0 SiS m.
for which
Assuming otherR(E) $. fi.
n
13
b~'H e T-fi ,
If
a' eA-a
n
a €A
n n
for all
such that
aeA,
n
a' ~A-a.
n
n
then
a' €A-A
n
n
Since
by Lemma 1.
a' eA-A l'
n
n-
a' eA-a
n
then
and the definition of
B ,
n
and
b '6:a--,
n n-1
b '€B-B-b.
n
n
Thus
there exists
and so
n
n
for all
b €B -B
n
n
(a'n- 1,b')€R
n
b'eB
Hence there exists
•
-a--n- 1
n-
1.
aeA l'
n-
and
By Lemma 2
such that
a' €E
n-1 n-1
Since
by assumption, it follows that
a'n-1
€
E
n-1 - En-2
and
(a'n- l' b')
~ M.
n
We can now repeat the above srg\DJlent beginning with
cess terminates when we arrive finally at
quence
(3.14)
where
1 s i s n.
(3.15)
o sis
(3.16)
8
0e
S-A,
b~+l e T-fi,
n.
aO€E ;
O
The pro-
having constructed a se-
14
(3.17)
1
~
i
~
n.
Now (3.15) and (3.16) are restatements of (3.7) and (3.8) and by
Lemmas 3 and 4, therefore, (3.17) implies (3.9).
It follows that the se-
quence. (3.14) is an augmenting chain (in reverse order), contradicting the
hypothesis.
Em • S-A-Am'
Thus
R(E _ ) So B
f 1
i
the pair
(C,D)
for
with
1
~
i
~
C • A-Am ,
m,
and
R(E)c
m-
D • Bm
B.
m
Since
constitutes a sup-
port as required, and the proposition follows.
Our preceding results are summarized in
THEOREM 1.
A matching is maximum if and only if it does not admit an
augmenting chain.
PRooF:
The necessity of the condition is stated in Proposition 1.
If there
does not exist an augmenting chain, then the support guaranteed by Proposition 3 has rank equal to the cardinality of the matching, which together
with Proposition 2 establishes the maxima1ity of the matching.
CoRCUARY.
matching
PRa>F:
If a matching
4»M': A' ua + B' ub,
4»M: A+B
where
is not maximum, there exists a
A'. X, Ii'. H,
and
a~A,
b~B.
By Theorem 1, there exists an augmenting chain with respect to
and the matching
M'
H,
may be constructed as in the proof of Proposition 1.
The following theorem, an immediate consequence of our preceding resu1ts, provides a generalization of the Ki:Snig-Egerv4ry Theorem to geometric relations.
15
TI-EOREM
2.
The maxim1Dl1 cardinality of a matching in
is equal to the minimum rank of a support.
PRooF:
By Theorem 1 a maximum matching
(G(S), R, G' (T»
2
M satisfies the hypothesis of
Proposition 3, so there exists a support of rank
,,(M).
By Proposition 2,
this support is minimum.
4.
IEFICIENCY AND CRITICAL SETS.
The results of §3 may be applied directly to obtain an expression for
the cardinality of a maximum matching in a geometric relation
G'(T»,
(G(S), R,
from which a generalization of the Marriage Theorem of Hall and
Rado follows as a corollary.
Before establishing these results, however,
we consider in this section the notion of a deficieney function on subsets
of
S.
The concept was introduced
by Ore [8] for bipartite graphs, and
may be extended to geometric relations as follows.
For any subset
(4.1)
<5
Since
(A)
=
S-(~UA2)'
(S-A )U{S-A ),
2
1
A of
S,
define the deficieney
<5
(A)
of
A by
reS) - reS-A) - r'(R(A».
S-(A nA )
l 2
are, respectively, identical to
it follows from (2.6) and (3.5) that
0
(S-Al)n(s-~),
is lower semi-
modular:
(4.2)
The rank functions
o(A)
by
2
is finite for all
r, r'
As;S.
are finite by (2.2), so the deficiency
Since
<5
is integer-valued and botmded above
reS), there exists a maximum deficieney
It can be shown that Theorem 2 is equivalent to a result obtained independently by Edmonds [5].
16
(4.3)
n •
6(A).
mdX
As.S
Subsets
Since
A of
S
satisfying
15 (A) • n will be called
n?; 0. and n
0«(1). 0.
> 0
if and only if all critical sets are
non-empty.
An immediate consequence of (4.2) is
PRePOSITION
4.
A OA
l 2
If
Ap A
2
en ticat sets.
are critical sets in
G(S),
then
A uA ,
l 2
are critical sets.
If follows from Proposition 4 that the family of critical sets is
closed under finite unions and intersections, and thus forms a ring of
sets. "the open sets in this ring will be of particular importance below.
cocwsure
In investigating their structure, it is convenient to consider the
A ~ '"
A induced by the closure operator
operator
subset
A'" S-B.
a
S -
~
implies
A
....
Al
I~ A
G(S) •
For any
we define
'"
A
Clearly
A ..... A of
is a coclosure operator:
....
=
~.
B.
~
N
A £. A.
A
~
ID
A.
The coclosed sets are the open sets of
and infima in the lattice
M(S)
and
AI:: A2
G(S).
Suprema
of open sets are given by
(4.4)
If
(4.5)
while
(4.6)
A· S-B
reS-A)
is any subset of
•
reS-A),
A £. A implies by (3.1)
r'(R(A»
s r'(R(A».
S.
then
r(B)· r(B),
that is
17
From (4.5) and (4.6) we have
'"
o(A)
(4.7)
~
o(A).
Thus
5.
PROPOSITION
If
.....
A is a critical set, then A is a critical open set.
Propositions 4 and 5, together with (4.4), imply
PRoPOSITION
The critical open sets form a sublattice ~(S)
6.
lattice M(S)
of the
of open sets in G(S).
In the case where
G(S)
is a free geometry, all subsets of
S are
open, and the sub lattice M (5) of the Boolean algebra M(S) is a ring.
O
By a well-known result (see e.g. [3]), rings are characterized latticially
by the distributive property: every distributive lattice is isomorphic to
a ring of sets.
THEOREM
PROOF:
For arbitrary G(S) ,
3.
we have as analogue
The sublattice
M (5) of critical open sets is distributive.
O
By a theorem of Birkhoff [3], a lattice is distributive if and only
if it contains neither Ml
nor
~
(Figure 1) as a sublattice.
E
E
Figure 1
18
We observe that if either M
l
or M is a sublattice of
2
~ (S),
then
( 4.8)
R(AnB)
since AnB
•
R(D),
is a critical set,
--...I
D" AnB,
and equality must hold in (4.6)
whenever both sets are critical.
Suppose first that
~
is a sublattice of MO(S).
(4.9)
A u B .. A u C .. E,
(4.10)
-----'
AnB
Then by (4.4)
.. ~--A n C .. D.
If follows from (4.9) that
B-A" C-A,
R(B-C)
so B-C S AnB.
£
R(AnB)
c
R(AnB)
c
R(C).
Then by (4.8),
Thus
r'(R(B»
•
r'(R(C) u R(B-C»
.. r'(R(C»,
which implies, since
o(B) .. o(C) ..
n, that r(S-B)" r(S-C). But B > C
in M , hence in M(S), so in the lattice L(S)
l
and hence reS-B) < r(S-C), a contradiction.
If M2 is a sublattice of NO(S) ,
(4.11)
Au B
=
then
A u C .. B u C .. E,
of closed sets
S-B
<
S-C,
(4.12)
--
19
,.,.........,
AnC •
AnB •
By (4.11) we have
."..,.
BnC •
A-C" B-C
III
D.
E-C,
R(E-C)
so E-C S. AnB.
c
R(AnB)
c
R(AnB)
c
R(C).
Then from (4.8)
Thus
r'(R(E»
which implies as before that
...
r'(R(C) u R(E-C»
III
r' (R(C» ,
r(S-E)'" r(S-C) ,
a contradiction.
The proof
is complete.
The foregoing results may, of course, be applied to subsets of G'(T).
In particular, for the converse relation
6'
R'
to R,
a deficiency function
is defined on subsets of T by
6'(B)
...
r'(T) - r'(T-B) - r(R'(B»,
with corresponding maximum deficiency
n' ...
max 6'(B).
BsT
The critical sets and critical open sets in G'(T)
vious manner.
By our preceding results, the critical open sets of G'(T)
form a distributive sub1attice MQ(T)
in
G'(T).
are defined in the ob-
of the lattice M'(T)
of open sets
20
5.
MAxIMJM MATOUNGS.
As a consequence of Theorem 2, we may express the cardinality of a
maximum matching in
(G(S), Rt G' (T»
n
in terms of the maximum deficiency
max o{A)
...
A=. S
defined in §4.
The result, which generalizes a theorem of Ore [8] for bi-
partite graphs, is
THEOREM
G' (T»
(S.l)
PRooF:
4.
The cardinality of a maximum matching
(G{S), R,
is
v(M)'"
reS) - n.
By Theorem 2, it is sufficient to prove that
of a minimum support.
D ... R{S-C),
If
(C,D)
...
r(C) + r'(R(S-C»
...
min
i So S
...
(r{B)
+
r'(R(S-B»
min (r{S) - o(S-B».
B s.
S
Since by (4.7)
reS) - o(S-B)
B S. S,
S
reS) - o(S-B)
we may rewrite the equality above as
p{C,D)
...
min (r{S)-o{S-B)
B S. S
...
reS) -
max
B S. S
...
r(S)-n
is the rank
is a minimum support, then clearly
and
p(C,D)
for any
M in
reS) - n.
o{S-B)
21
The case
n
=0
in Theorem 4 provides a generalization of the Marriage
Theorem of P. Hall and RaOO to geometric relations:
CoROLLARY.
R, G' (T»
(5.2)
There exists a matching of cardinality
reS)
in
(G{S),
if and only if
reS) - r{S-A)
S
for all subsets A of
r'{R{A»
S.
Recalling the definition (2.7) of the rank function for a minor. we
observe that condition (5.2) may be alternatively stated: as follows:
For every subset
ASS,
the rank of the contraction of G{S)
does not exceed the rank of the reduction of G'{T)
(5.3)
s
r[S-A,S] (A)
to
R{A},
to A
i.e.
r'[~,R(A)](R(A».
We close this section by characterizing the minimal cri tical set, which
by (4.7) is also the minimal element in the lattice MO{S).
PROPOSITION
7.
Let
ae:S belongs to
Aa
by one when G{S)
n
Aa
be the minimal critical set in S.
if and only if the maximum deficiency
is contracted to
S-8'.
remains unchanged when contracting G{S}
PRooF:
The deficiency
G' (T»
is
(5.4)
o[a,s]{A)
o[8,S]{A)
a~AO'
If
to
Then a point
n
is reduced
the maximum deficiency
S-8.
of a set A S S-a for
D
r[a,s]{s-a) - r[a,s]{S-(Aua»
•
(r(S)-r{a»
•
o{A}.
(G[a,S]' R,
- r'(R(A»
- (r(S-A)-r(a}) - r'(R(A»
22
a~AO'
afS-AO' so a £ S-AO since S-AO is closed. Thus a~AO
implies A £ S-a and hence by (5.4) (Gra,s]' R, G'(T» has maximum deO
ficiency n.
If
then
If
aE:A ' however, it follows from (5.4) and the minimal1ty of AO
O
that the maximum deficiency is reduced when G(S) is contracted to 5-a.
It is sufficient to show that the set AO-a has deficiency n-1
R, G' (T».
Now since
6
But
in
(G(S),
a £ A.a,
(Au-a) •
reS) - r«S-Ao)ua) - r'(R(A.a-a»
•
reS) - r(S-An) - 1 - r'(R(Ao-a»
~
reS) - r(S-Ao) - r'(R(An»
CI
HA ) - 1
...
n - 1.
o(Ao-a) ~ n-l since
- 1
O
An
is minimal, so equality holds, and the proof
is complete.
6. THE
STRUCTURE OF HI NIM SUPPORTS.
Let L(S), L'(T)
G' (T),
respectively.
closed set of
G(S)
ane similarly for
denote the geometric lattices of closed sets in G(S),
The same symbol will be used to denote an arbitrary
when it is regarded as an element of the lattice L(S),
G'(T).
Consider now a pair of functions
a: L(S) + L'(T),
defined by
(6.1)
a(C)
•
R(S-C),
a'(D)
•
R'(T-D),
a': L'(T) + L(S)
23
where
R' S TxS
is the converse relation to
R.
Clearly
cr, cr'
are order-
inverting
(6.2)
Furthermore, we have for any
(6.3)
cr'cr(C) S C,
Ct:L(S),
DE:L' (T)
crcr'(D) S D,
as is easily verified.
We deduce from (6.2) and (6.3) that the pair of functions
a dual Galois connection between the lattices
L(S)
a Galois connection between their dual lattices.
of Galois connections
crcr'
Q(S), Q'(T)
with the restriction of
cr
to
Furthermore, for any subset
and similarly for
L' (T),
Q' (T)
and
L(5), L' (T) ,
as follows.
no pair
CuD,
(Cl,D )
1
cr 'cr ,
of coclosed elements are anti-isomorphic,
Q(S)
{C : it:I}
i
providing a canonical anti-isomorphism.
of
Q,
L' (T) •
Let us define a support
of closed sets, such that
is a support.
that is,
respectively, and that the
The preceding observations may be related to supports of
G'(T»
forms
It follows from the theory
(cf. Ore [7]) that the composite functions
are coclosure operators on
quotient lattices
and
cr, cr'
(C,D)
ClUD
l
(G(S), R,
to be irredundant if
is a proper subset of
Clearly any minimum support is irredundant.
24
8.
PROPOSITION
A pair
(C,D)
of closed sets in G(S), G'(T)
red'mdant support if and only if
D!Q'(T),
PRooF:
(C,D)
D ~ R(S-C)
By symmetry,
IS
(C,D)
is an irred\nldant support.
a(C),
C" a'(D),
Conversely, if
(C,D I )
or equivalently
C" a'(D).
Suppose
support,
D D a(C),
C!Q(S),
is an ir-
C. a'a(D),
D. a(C),
so
then
C!Q(S).
(C,D)
is a support, but
is not a support for any proper closed subset
DIeD.
is not irredundant, there exists a closed set C
I
= a'(D).
CI 2 R'(T-D)
D D a(C).
and irredundancy implies that
i.e.
C!Q(S),
By the definition of a
a'a(C)
But then in L(S),
<
C,
Hence if
¥C
such that
which contradicts
C!Q(S) •
Consider now the minimum supports.
consisting of those flats in
then LO(S) S Q(S)
S Q'(T)
G(S), which are elements of minimum supports,
under a.
is order-anti-isomorphic to LO(S)
C
a'(D).
D
is the subset of L(S)
by Proposition 8,and the corresponding subset LO(T)
support is of the form
D€LO(T),
If La (S)
(C,D),
where
Thus every minimum
D = a(C),
C€LO(S),
The structure of the subsets
LO(S) ,
or equivalently,
LO(T)
will
become apparent from our results in §4 through
PROPOSITION
9.
A
closed set
C in G(S)
port if and only if its complement
similarly for closed sets
PRooF:
Suppose
D in
S-C
is an element of a minimum sup-
is a critical open set, and
G'(T).
S-C is a critical open set.
p(C,a(C»
Then
..
r(C) + r'(R(S-C»
•
r(S) -
<5 (S-C)
.. r(S) - n,
so
(C,a(C»
is a minimum support by Theorems 2 and 4.
25
Conversely, if
is a minimum support, then
(C,a(C»
6(S-C)
a
reS) - r(C) - r'(R(S-C»
•
reS) - p(C,a(C»
CI
n,
sO S-C is a critical open set.
The same result holds for a closed se t
The quotients
Q(S), Q'(T)
with suprema in L(S), L'(T),
D in G' (T)
by symmet ry •
are lattices in which suprema coincide
respectively.
For the subsets
LO(S), LO(T),
the result is considerably stronger.
PROPOSITION
10.
The subsets L (S), L (T) are anti-isomorphic, distriO
O
butive sublattices of L(S), Lt(T), respectively, and the restriction of
to LO(S) provides a canonical anti-isomorphism.
PROOF: By Theorem 3 the critical open sets in G(S), G'(T)
a
butive sub lattices of the lattices M(S), M'(T)
form distri-
of open sets, respectively.
Since distributivity is preserved under dualization, the complements of
critical open sets in G(S), G'(T)
L'(T).
G'(T)
By Proposition 9 the complements of critical open sets in
are precisely the elements of LO(S), LO{T).
isomorphism from LO(S)
to LO{T)
theory, since LO(S) £ Q(S)
The anti-isomorphism
set
form distributive sublattices of L{S),
{C : i£I}
i
a
is an anti-
follows from the Galois connection
and LO(T)
0
That
G{S),
from LO(S)
is the image of LO(S)
to LO(T)
of lattice elements in LO(S),
under o.
implies that for any
26
where suprema and infima are as in L(S), L'(T), since LO(S), LO(T)
sub lattices •
are
Our results on the structure of minimum supports are summarized
in
THEOREM
5,
Let
M (S), M (T)
O
O
be the sublattices of M(S), M' (T)
whose elements are the critical open sets in
Let
LO(S), LO(T)
G(S), G'(T),
be the sublattices of L(5), L'(T)
respectively.
whose elements
are the set-theoretic complements of the elements of M (S), M (T) • Then
O
O
the sub lattices LO(S), L'(T) are distributive and anti-isomorphic. The
minimum supports of
(G(S), R, G'(T»
consist of all pairs
(C, a(C»,
and for any set
{(Ci,a(C
i
»:
i€I}
of minimum supports, the pairs
are minimum supports.
We conclude with an esample illustrating the foregoing theory.
the geometric relation consisting of the triple
G(S)
is an arbitrary open pregeometry and
1
(G(S), 1, G(S»,
Take
where
the identity relation.
Since in this case the maximum cardinality of a matching obviously equals
r(S), we have
n· 0
by Theorem 4.
Hence the critical sets
A£S
are
those which satisfy
(6.4)
NOl'1
rCA) + reS-A)
•
reS).
it is easy to see that any such set
open as well.
A is closed, and hence by symmetry
Thus the lattices of critical and critical open sets coincide
27
and, since according to (6.4) this lattice is uniquely complemented,
it is by Theorem 3 a Boolean algebra.
The dual lattice
also a Boolean algebra and every minimum support of
the form
(C, S-C),
C satisfying (6.4).
L (S)
O
is hence
(G(S), I, G(S»
is of
Since (6.4) Characterizes the
separators of an open pregeometry, we thus obtain the well-known result
(see e.g. [4]) that the separators of a geometric lattice
algebra which is a sublattice of
L form a Boolean
L.
ACKNOWLEDGMENT
It is a pleasure to record our gratitude to Professor G.-C. Rota who
suggested much of this work and kept an encouraging interest throughout.
REFERENCES
[1]
M. Aigner and T .A. Dowling, Matching Theorems for Combinatorial
Geometries, Bull. Amer. Math. Soa.~ 76 (1970), 57-60.
[2]
C. Berge, 'the Theory of Graphs and its App'ti(Jations~ Dunod, Paris,
1958. English translation N:rtH U€ll.: London and Wiley, New York,
1962.
[3]
G. Birkhoff, Lattice Theory, A. M. S. Colloq. Publ.,
25, 1940, 1948,
1967.
[4]
H.B. Crapo and G.-C. Rota, On the Foundations of Combinatorial Theory:
Combinatorial Geometries, MIT Press, Cambridge, Mass. 1970.
[5]
J. Edmonds, Submodular functions, matroids, and certain polyhedra,
Proaeedings of Calgary Inter. Conf. on Corrbinatorial
Gordon and Breach, Inc., New York 1970.
[6]
StrouatUX'es~
L. Mirsky and H. Perfect, Applications of the notion of independence
to problems of combinatorial analysis, J. Combinatorial Theory~
2 (1967), 327-357.
[7]
o.
Ore, Galois connexions, Trans.
Am. Math. Soa., 55 (1944), 493-513.
28
[ 8]
o.
[9]
R. Rado, A theorem on independence relations, Quart. J. Math.
O%ford Ser., 13 (1942), 83-89.
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