The theory of convex geometries - Springer Link

PAUL H . EDELMAN * AND ROBERT E . JAMISON **

THE THEORY OF CONVEX GEOMETRIES

1 . INTRODUCTION

The purpose of this paper is to develop the foundations of a combinatorial abstraction of convexity that we call convex geometries. This work has gone on for several years ([14]-[16], [27]-[32]), and this paper will serve to present the basic aspects of the theory . Some of the results have appeared before . Others have been previously stated without proof and still others are new. (The authors have previously referred to these objects by the cacophonous name ` antimatroids' . We hope there is time to rectify this and that Gresham's Law does not apply to mathematical nomenclature .) Our approach to abstract convexity has many advantages . It is broad enough to encompass all the standard examples as well as many less standard ones . There are many natural, equivalent ways of defining convex geometries . The lattice of convex sets can be completely characterized . Our axiomatization allows one to apply techniques of graph theory and ordered set theory as well as techniques from standard convexity theory . Recently it has been discovered that convex geometries are dual to an important class of greedoids . The rest of the paper is organized in five parts . The next section contains the definitions of a convex geometry and proves various equivalences . Section 3 contains a discussion of the important examples of convex geometries . Of particular interest in this section are some examples related to k-families of a poset . In Section 4 we characterize the lattice of convex sets of a convex geometry and apply techniques from lattice theory to gain additional information. Section 5 presents the important operations on a convex geometry . This leads to defining a lattice structure on the collection of all convex geometries on a fixed ground set . We also prove a Dilworth-type inequality, Theorem 5 .3, in this section . The last section, Section 6, has some final remarks and a discussion of some open problems . Throughout this paper we will concern ourselves solely with finite sets . If X is a finite set then X denotes its cardinality . If P is a poset then P* will denote its order dual . If x, y c P then [x, y] refers to the interval I

I

* Supported in part by NSF Grant MCS-8301089 . ** Supported in part by NSF Grant ISP-8011451 (EPSCoR) . Geometriae Dedicata 19 (1985) 247-270. 0046-5755/85.15 . © 1985 by D . Reidel Publishing Company .

248

PAUL H . EDELMAN AND ROBERT E . JAMISON

x 5 z 5 y} in P with the induced order . The set { 1, 2, . . ., n} will be denoted [n] . {z I

2 . BASIC EQUIVALENCES In this section we define convex geometries and discuss their basic properties . We also introduce various equivalent characterizations. Let X be a finite set and 2' a collection of subsets of X with the properties (Al) (A2)

Q A

and

E 2' E 2'

and

X B

E 2'

E 22

implies A n B

E

2' .

2' is called an alignment of X . We can alternatively think of 2' as being a closure operator . That is, for any subset A of X we define the closure of A, 2'(A) to be It(A) =

n

(Cs2'IC2A)

c.

It is easy to check that 2' is a closure operator on X, i .e . 2' is a function from 2x to itself such that (C1)

A c 2'(A)

(C2)

A

(C3)

2'(2'(A)) = 2'(A)

c

B

implies

2'(A) ~ 2'(B)

with the additional condition that 2'(Q) = 0. The subsets in 2' or, equivalently, those subsets of X of the form 2'(A) for some A X, will be called convex sets. Strictly speaking, it is not necessary to include the assumption that 0 E 2' but it simplifies matters to make it . We will move between these two interpretations of 2' as is convenient . The purpose of this section is to prove various combinatorial abstractions of convexity are equivalent . We begin with a few standard definitions. Let A be a subset of X . A basis for A is a minimal subset S c A such that 2'(S) = 2'(A). A priori there may be many bases for a particular set . A point p e A is called an extreme point of A if p ~ 2'(A - p) . The set of extreme points of A is denoted ex(A) . Extreme points may or may not exist. Notice that ex(A) is contained in every basis of A . Suppose p e X . A copoint C attached at p is a maximal convex set in X - p. There may be more than one copoint attached at a point . c


249

Finally we define the anti-exchange condition . We say that Y is antiexchange if given any convex set K, and two unequal points p and q in X, neither in K, then q c 2(K u p) implies that p ~ 2'(K u q) . We are now ready to prove the main result of this section . Some of these equivalences have appeared elsewhere [6], [36] . We include all the proofs for completeness . THEOREM 2 .1 . Let Y be an alignment of a set X . Then the following are

equivalent : (a) 2' is anti-exchange . (b) For every convex set K, there exists a point p e X such that K u p is convex . (c) For every point p and C a copoint attached at p, C u p is convex . (d) Every subset A s X has a unique basis . (e) For every convex set K, K = 2'(ex(K)) . (f) For every convex set K and p 0 K, p e ex(2'(K u p)) . Proof. Our proof consists of two cyclic implications : a=> b=> c=> a

and a' d=> er .f=a . I . a => b . We shall show that for any convex sets C and K, K !i~ C, there exists a point p e C - K such that K u p e Y. This will be done by inductionon IC-KI .If IC-KI = I then K u p = C e T . For q c C - K, if 22(K u p)=#C then we have K t 9'(K u p) f C and we are done by induction using K and 2'(K u p) . If 22(K u p) = C then for any q e C - K, q =~ p, we have 22(K u q) f 2'(K u p) by the antiexchange property and so K !i~ 2'(K u q) !~ C and again we are done by induction . II . b => c . Let C be a copoint attached at p. By (b) there exists a point q such that C u q is convex . Since C is a maximal convex set in X - p, q must equal p . III . c ==~> a . Suppose K is a convex set with p and q points such that p =~ q, p, q 0 K and q e 2'(K u p). Let K' be a copoint attached at q which contains K . Then q e 2'(K' u p) . Moreover, p 0 K', or else q e 2'(K') = K' . Since by (c) 2'(K' u y) = K' u y we know that p ~ 2'(K' u q) and so p Y(K u q). IV . a => d. Suppose K is a set with two bases, B 1 and B 2 . Let p e B 1 , and B'2 be a minimal subset of B 2 such that B 1 - p u B'2 spans K. If q e B2

250


then B 1 - p u B'2 - q is not a spanning set . Let C = 2'(B 1 - p u B2 - q). Then 2'(C u p) = 2'(C u q) = 2'(K) . So p c 2'(C u q) and q e 2'(C u p) which implies that p = q by' (a). Hence every element of B 1 is in B 2 . Similarly, one can show that every element of B 2 is in B 1 and hence B 1 = B 2 . V . d ==> e . We shall show that ex(K) is the unique basis for K . It is clear that ex(K) is contained in any basis for K . Suppose p e K and p ~ ex(K) . Then p e 2'(K - p) and hence there is some basis for K not containing p . Since the basis is unique by (d), the basis for K must be ex(K) . VI . e =>f Suppose K is convex and p ~ K . Define a set S by S = S(K u p) - (K u p) . It is clear that S n ex(2'(K u p)) _ 0 . But since ex(1(K u p)) spans 1(K u p) by (e), we have p e ex(It(K u p)) . VII. f = a. Let K be a convex set and p =~ q, p, q ~ K . Suppose q e 2'(K u p) . By (f), p e ex(2'(K u p)). Hence p 0 2'((K u p) - p) . This implies that 22(K u q) !~ Y(2(K u p) - p) and sop ~ 2'(K u q) .

0

We shall call the pair (X, 2') satisfying the conditions of Theorem 2 .1 a

convex geometry (c .g .) . If all the points of X are closed, the convex geometry will be called atomic . Frequently we will refer to 2' as a c .g . if the ground

set X is unambiguous . Notice that in the proof of a implies b in Theorem 2.1, we show something stronger. If C and K are convex sets and K is minimal with the property that C f K then K = C u p for some p e X. Thus we have shown the equivalence THEOREM 2 .2 . The pair (X, 2') is a convex geometry if and only if every maximal chain of convex sets of 0!iE C15C2 has the same length I X I

...

X

.

0

Another useful equivalence concerns a relation one can put on X . Let K be a convex set and define the relation < K On X - K by p< K q

if and only if p e 2'(K u q) .

This relation will be called the K -factor relation . THEOREM 2 .3 . (X, 2') is a convex geometry if and only if the K factor relation is a partial order on X - K for all convex K .


251

Proof. The K-factor relation is reflexive and transitive for any alignment .

The antisymmetry is easily seen to be equivalent to the anti-exchange property . El

The reader should note that the K-factor relation is an equivalence relation for all convex K, if and only if (X, 9) is a matroid . In the next section we shall use the K-factor relation to characterize certain convex geometries . Finally in this section we shall prove a convex geometry analogue of a result of Greene [19] for matroids . First we prove another equivalence.

THEOREM 2.4. (X, Y) is a convex geometry if and only if every copoint is attached at a unique point . Proof. Let C be a copoint attached at two points p and q . If K is a convex set such that C !~ K, then p and q are both contained in K . This contradicts condition (b) of Theorem 2 .1 . On the other hand, suppose (X, 2) is an alignment such that every copoint has a unique point of attachment . Let C be a copoint attached at p . For every q, q E X - (C u p) there is a copoint Cq attached at q such that C !4 Cq . Notice that p e Cq for all q and hence

n

qcX-(Cup)

Cq =C u p

since for each q E X - (C u p), q o C q . Thus C u p is convex and condition (c) of Theorem 2 .1 is satisfied . 1] Let Val(X, Y) be the number of copoints in the alignment (X, Y) . Then

we have

COROLLARY 2 .5 . If (X, 2) is convex geometry then Val(X,') > I X 1 . 3 . EXAMPLES In this section we present the principal examples of convex geometries and discuss some of their properties . Some of these have appeared elsewhere [34], [36] . Of particular interest are the examples related to k-families of a poset, in Example II below, which to our knowledge have not appeared elsewhere . We also use this section to define some terms which will be needed in subsequent sections .

252


Example I : Points in R" and Oriented Matroids

Let X = {XI, x2, . subset of X define

. ., x k }

be a set of points in R" . For A = {a l , a 2 , . . .,

Y(A)={xeXIx= Y~tat ~j%0,YAi=1} .

In other words, .(A) is the intersection of the convex hull of A with the set X . The Krein-Milman theorem and Theorem 2 .1(e) imply that (X, 2) is a convex geometry. This situation has a natural generalization to oriented matroids . This was presented in the language of Bland and Las Vergnas [8] in [14] . It is based upon a closure operator defined by Las Vergnas [38] . See also the closure operator h defined in [18, p . 204] . We give a brief outline of this construction . For details see [38] and [14] . Let M = (E, (9) be an acyclic oriented geometry, i .e . an acyclic oriented matroid with no two point circuits . Let A be a subset of the points E of M and define (*)

Y M(A) = A u {x e E - A I there is a signed circuit X

of M such that X - = {x} and X + ( A} THEOREM 3 .1 . Let M = (E, (E, ' M) is a convex geometry. Proof. See [14, Th. 2.1] .

(2) be an acyclic oriented geometry. Then

There are two special cases of this example to which we wish to call attention . Goodman and Pollack [20, Def. 1 .1] consider a notion of convexity on arrangements of pseudo-lines in p2 and prove various analogues of classical convexity theorems. As can be seen using the work of Folkman and Lawrence [18, §IV] on pseudo-hemisphere arrangements, Goodman and Pollack's definition of convexity is exactly the same as that given above for acyclically oriented matroids and hence is a convex geometry . See [9] for details of the relationship . Using Goodman and Pollack's correspondence between pseudo-line arrangements and allowable sequences [22] the structure of a convex geometry is induced on those sequences as well [21, Def. 2 .9] .


253

One other special case deserves mention . Let G be an acyclically oriented simple graph with directed edge set E . Then the closure (*) in this special case reduces to P G(A) = A u {e c- E I e is in the transitive closure

of some subset S c A}

for all A c E. Suppose G is a comparability graph for some poset P . Then the convex sets of P correspond to subposets of P, i.e . a partial order on the same set as P but with a subset of the relations of P . This convex geometry can be used to study various aspects of P . See [43] for details . Example II : Ordered Sets

Let P be a partially ordered set . Let A be a subset of the points of P and define (j)

-9 p(A) = {x c P I x< a for some a e A} .

We shall call this the downset alignment on P. It is easy to see that this produces a convex geometry . The convex sets of -9 P are the order ideals (down sets) of P . If K is convex then ex(K) is the set of maximal points in K. The 0-factor relation is isomorphic to P . In fact convex geometries that arise in this way are easily characterized by THEOREM 3 .2 . The convex geometry (X, °) arises from the downset

alignment on a poset P if and only if It(A u B) = 2(A) u 1(B) for all sets A,B-X.

Proof. It is clear that the downset alignment on a poset satisfies the above property . Conversely, let (X, Y) be a convex geometry satisfying the property that ./(A u B) = T(A) u -T(B) for all sets A and B . Partially order X by the 0-factor relation, i .e . x < y if and only if x c Y(y) . We shall show that -T is the downset alignment on X under this partial ordering . Suppose K = {k 1 , k,, . . ., ke } is convex . Then K ='(K)

= U

2 (ki) = { y I y , k + 1 . rk,k+1(X) being connected as equivalent to showing that for each set of adjacent ranks of a meet-distributive lattice, the edges of the Hasse diagram form a connected graph . This property is implied by the fact that L is Cohen-Macaulay . See [3, Th . 6 .4] and [42, Th. 4.3] . El 1

We wish to thank A . Bjorner for pointing out the relevance of [11] .

262

PAUL H . EDELMAN AND ROBERT E . JAMISON 5 . CONSTRUCTIONS

In this section we discuss the principle constructions and operations for convex geometries . We shall also discuss some of their uses . Other constructions will be mentioned in Section 6 . 1 . Joins Given two alignments on the same set (X, 2') and (X, A') we define the join of 2' and A#, denoted 2 V ~f, to be the alignment 2'VA'={CgX1C=LnMforsome Le2andMeA'} . The following theorem is proven in [15, Th . 2 .2] . See also [30, p . 135] . THEOREM 5 .1 . If (X, 2') and (X, Ad) (X, 2'V /d) is also a convex geometry .

are convex geometries then El

Let CG(X) be the set of all convex geometries on X . Partially order CG(X) by (X, 2') < (X, mil) if and only if 2 c A' . Then Theorem 5 .1 shows that CG(X) under this partial order is a join semilattice . Let E be a total ordering of the set X . Then it is easy to see that the monotone alignment on E, -9E, is a minimal element in CG(X) since iE contains exactly one subset of X of each cardinality . We shall show that the only join-irreducibles of CG(X) are of this form . Let (X, 2') be a convex geometry . A compatible ordering of 2' is a total ordering of the element of X, x 1 < x 2 < . . . < x„ such that the set Ci = {x 1 , X2, . . . , x i } e 2' for all i, 1 < i < n . Note that a compatible ordering of 2' corresponds exactly to a maximal chain in L, i .e . 0 c C 1 s-= C, c • • • c C„ = X is a maximal chain in L, and every maximal chain in L 2 gives rise to a compatible ordering . Denote by Comp(Y) the set of compatible orderings of Y . THEOREM 5 .2. For every convex geometry (X, 2')

V

we have

2' _

9E E E Comp(.T) -

Proof. Let A- = V E E Comp(Y) -9 E . We first show that if C e 2' then C E I' . Take a maximal chain in Ly containing C and let E be the corresponding compatible order. Then C e -9E and hence C e A' . Conversely, if C e .%'' then C = n E E Comp(.') K E where KE E -E . Since K E e 2' for all E, and 2' is closed under intersection we have C = E 2' . nE E Comp(2) KE

El


263

Theorem 5 .2 shows that the monotone alignments form the building blocks of all convex geometries. It also prompts the question of what is the fewest number of monotone alignments needed to generate a particular convex geometry. A set of linear extensions E 1 , E2 , . . ., EK is said to realize 2' if 2' = Uk=1 .9E, . Let cdim(2'), the convex dimension of 2', be the minimum number of compatible orderings needed to realize Y . Let d(2') be the cardinality of a maximum-sized independent set in Y . THEOREM 5 .3 . For 2' a convex geometry we have cdim(2') >, d(2') . Proof. Let I = { i1, '21 . . . . ik } be a maximum sized independent set in Y . Suppose that 2' is realized by the orderings E 1 , E 2 , . . ., E r . Since I is an independent set we know that i3 0 2'(I - is) for all 1 < s S k. Hence for each s, there must be some compatible ordering Ej such that in Ej is is larger than all the other elements of I . Hence r > 111 . El

kVCA$ [1,2,31

[1,31 [3]

[1,2 .41

.

[1 .41 [4]

0

Fig . 1

It is natural to ask when the inequality in Theorem 5 .3 is in fact an equality . By applying Dilworth's theorem (see for instance [1, p . 400]) we have that cdim(-9 P) = d(-9P ) for any poset P . Unfortunately equality does not always hold . The alignment whose lattice of closed sets is depicted in Figure 1 has convex dimension 3 even though its maximum sized independent set has size 2. This example is due to Saks [40] . We now return to the semilattice CG(X) . THEOREM 5 .4. The length of all maximal chains in an interval [Y, AC] in CG(X) is I A - I - I P I . Proof. We shall show that given two c .g.s on X, 2' and A', where 2 5 V- in CG(X), that there exists a convex set C E 1' - 2' such that L u {C} is also a e .g. The theorem then follows easily . El

264 Let A

PAUL H . EDELMAN AND ROBERT E . JAMISON c'

be defined by

A={KEG-2'I]CE2' where C= K u {x} for some x e X} . That is, A is the collection of convex sets in 1 not in 2' such that they are covered by some convex set in Y . The collection A is nonempty since the co nvex set initC - 2' of largest cardinality must be in it . Let C E A be a set of minimum cardinality in A . We claim that 2' u {C} is also a e .g. First we show that T u {C} is closed under intersection . Suppose C' n C 0 2' for some C' E Y . Since 2 c i, we know that C' n C E _f . Moreover, since C u {x} E 2' for some x e X we know that C' n (C u {x}) E -T . If x 0 C' then C' n (C u {x}) = C' n C, which is a contradiction . On the other hand, if x E C' then C' n (C u {x}) = (C' n C) u {x} and hence C' n C E A, which contradicts the choice of C as an element of A of minimum cardinality . Hence, C' n C E 2' for all C' E 2', and thus 22 u {C} is closed under intersection . 2' u {C} is closed under intersection and contains both 0 and X . By the choice of C we have ensured that 22 u {C} satisfies condition (b) of Theorem 2.1 . Hence 2' u {C} is a convex geometry . COROLLARY 5 .5 . CG(X) is a ranked join-semilattice with rank function rk L' = 12 I - I X I - 1 . Proof. The minimal elements of CG(X) are monotone alignments which have size I X I + 1 . The corollary then follows from Theorem 5 .4 . E] defined by

Using the construction in the proof of Theorem 5 .4 we can discuss a structural property of the set Comp(2') . Let G((B x) be the graph whose vertices are the permutations of X where two permutations are adjacent when they differ by an adjacent transposition . This graph is the one-skeleton of the permutohedron . 2 We identify a permutation x 1 x 2 . . . x, of X with the linear ordering x 1 < x 2 < . . . < x,, . The following theorem was independently proven in [37, Th . 3.2] . Our proofs are considerably different . THEOREM 5 .6.

For every convex geometry (X, 2'), Comp(Y) induces a

connected sub graph in G((B x ) .

Proof. We prove this theorem by induction on the rank of 2' in CG(X) . Certainly the theorem is true if 2' is a monotone alignment, i .e . rk(2) = 0. Suppose rk(2') = k. Then by Theorem 5 .4 we know that 2' covers a c .g . 2', 2

See Berge [44, page 135] .


265

rk(2') = k - 1, and so 2 = 2' u {K} . By induction Comp(2') induces a connected subgraph in G(S X). The only compatible orderings of 2' that are not compatible for 2' correspond to maximal chains in L 2 that contain K . Consider such a chain c = 0 f K 1 ~-, • • • f K" s-, K t K t . . . + X whose corresponding compatible order is E = x 1 < . . . < y < y' < . . . < x„ where K = K" u y and K' = K u y' . Since both K" and K' are in 2' there is some convex set K* E .' such that K" c K* c K' and hence K* = K" u y' and K' = K* u y . Thus the compatible ordering corresponding to the chain c*=0 s~ K 1 $~ • • • 5~ K"ciK*c,- K'f • • • ~,- X inY'isE*=x 1 < . . . < y' < y < . . . < x„ which is adjacent to E in G( (Z x ) to one that is in Comp(') . Hence, by induction, the induced subgraph of Comp(Y) is connected . 0 We now turn to a property of total order alignments and compatible orderings of a poset. THEOREM 5 .7 . WP

For P a poset we have

= V

WE

where the join is over all compatible orderings

E

of P .

Before we prove this theorem we will need a lemma LEMMA 5 .8 .

Let P be a poset, C

E

16, and x e P, x

a compatible ordering T of P such that x ~' T(C) .

0

C . Then there exists

The proof is by induction on P . The convex set C cannot contain all the maximal and minimal elements of P or else C = P. Without loss of generality let m c P be a minimal element such that m 0 C . If m = x then any compatible ordering with x as the minimum element will work for the lemma . If m ~= x, let E' be a compatible ordering of P - m such that x ~ CE .(C), which exists by induction . Then the compatible ordering gotten from E' by adding m first satisfies the lemma . Proof of lemma .

Proof of Theorem 5 .7 . Let _'f = V 'E where the join is over all compatible orderings E or P. Suppose C E Wp . Let D = n 'E (C) where the intersection is over all compatible orderings E of P . We claim that C = D . Certainly C c D since C c W E(C) for all E . If x 0 C then by the lemma there is a compatible ordering E' such that x ~' E'(C) . So C = D and 16, c . We now show that 'E 91 W 1 for any compatible ordering E of P. It then follows that .( c 16, and we shall be done . Suppose A E WE and A 0 Wp .

266


Then we can find three distinct points x, y, and z in P such that x < z < y in P, x and y in A, and z not in A . Since E is a compatible ordering of P,

x < z < y in E as well and since A C_ WE we know that z e A, which contradicts the choice of z . Thus A C_ WE implies that A E W, and so W E c W1 . 0 Theorem 5 .7 leads us to the following definition : given a poset P let the interval dimension of P, idim(P), be the minimum number of total order alignments whose join is WP . The interval dimension of P is different than the dimension of P in the sense of Dushnik and Miller [12] as one can see by looking at the example of an antichain . The dimension of an antichain of size n is 2 whereas its interval dimension is the least integer greater than n/2 . 2 . Minors In this section we show that the standard construction of minors as done in matroid theory also works in the case of convex geometries . Let (X, 2') be an alignment and Y a subset of X . Define the relative alignment on Y, or the restriction of 2' to Y to be the alignment on Y 11Y={Cn YICa2'} . THEOREM 5 .9 . If (X, 2') is a convex geometry and Y c X then (Y, 2° I Y) is also a convex geometry . Proof. Let C E 2 I Y and a, b e Y, unequal and neither in C. Suppose a E 2'I Y(C u b) and b e 2'I Y(C u a) . Then by the definition of the relative alignment we have a e 2'(C u b) n Y which implies that a e 2'(C u b) and similarly b c 2'(C u a) which contradicts that (X, 2') is a convex geometry. Hence 2' I Y is anti-exchange and so (Y, 2'I Y) is a convex geometry .

0

Again let 2' be an alignment of a set X and Y c X, Y E Y. Define the contraction of 2' with respect to Y, 2'/Y, to be the alignment on X - Y defined by

2'1Y={Cs~

X-YIC=2'(D u Y)-Y forsome D~X-Y} .

(The assumption that Y e 2' ensures that

0

e Y/Y .)

THEOREM 5 .10 . If (X, 2') is a convex geometry and Y ~ X, Y e 2', then (X - Y, 2'/Y) is also a convex geometry .


267

Proof. Let C E 2'/Y and a, b e X - Y, unequal and neither in C . From the definition of 58/Y we know that C = 58(D u Y) - Y for some D e Y . Let K = C u Y so that K E Y . Suppose a e Y/Y(C u b) and b e Y/Y(C u a) . Then a e 58(K u b) and also b e 58(K u a) which contradicts that (X, 58) is a convex geometry . Hence (X - Y, I/Y) is also a convex geometry . F1 A minor of an alignment (X, T) is any alignment of a subset Y of X obtained by a sequence of restrictions and contractions . So COROLLARY 5 .11 . Every minor of a convex geometry is a convex geometry . LI We are now able to present a forbidden minor characterization of convex geometries. Let Q 0(2) be the alignment on the two element set [2] where Q 0 (2) = { 0, {1, 2}} . It is clear that ([2], Q 0 (2)) is not a convex geometry. THEOREM 5 .12. An alignment (X, 58) is a convex geometry if and only if it has no minor isomorphic to ([2], Q0 (2)) .

Proof. It follows from Corollary 5 .5 that if (X, 58) has a minor isomorphic to ([2], Q 0(2)) then (X, 58) is not a convex geometry . Suppose (X, T) is not a convex geometry. We shall show that some minor of 58 is isomorphic to ([2], Q 0 (2)) . Since (X, 58) is not a convex geometry we can find a convex set C E 58 and two distinct points a and b in X but not in C such that a e 58(C u b) and b e 58(C u a) . Consider the

minor obtained from 58 by contracting by C and then restricting to {a, b} . This minor has only two convex sets, the empty set and {a, b} and hence is isomorphic to ([2], Q 0(2)) . 0 6 . FINAL REMARKS AND OPEN PROBLEMS In this section we discuss some of the topics which we have not covered in detail . We shall also mention some of the important open problems concerning convex geometries .

A major topic we have chosen to exclude is the evaluation of various parameters associated with an alignment . We have not touched upon the Caratheodory number, Radon number, or any other parameter with the exception of the Helly number . There is, at the moment, little theory applicable to all convex geometries and what is known is restricted to an example by example case. The interested reader can consult [20], [27]-[29], [9] and various references in [30] .

268


The second topic which we have not covered in detail is the characterizations of types of convex geometries . These problems tend to be very difficult and require intricate proofs and special constructions . For the characterization of convex geometries arising from the order convex alignment on a poset, see [31] . For the characterization of those arising from trees and block graphs, see [28] . Finally we mention what we feel are the most important open questions related to convex geometries . Problem 1 . Characterize those convex geometries arising from the convex alignment on a finite number of points in R" . It may be easier to characterize those arising from convexity on an acyclic oriented matroid . Problem 2. What convex geometries can be written as the join of total order alignments? It is known that the order convex alignments on posets and the convex alignment on points in R" can be so expressed but there are others as well . Problem 3 . Develop a theory of convex dimension for convex geometries . When does equality hold in Theorem 5 .3? [Note added in proof: Much of this theory has been worked out by the first author with Saks . We still do not know when equality holds however .] Problem 4 . What collections of total orders can be obtained as Comp() for some Y a convex geometry? Theorem 5 .6 provides a necessary condition but it is not sufficient . Problem 5 . What is the convex geometry associated with s?k (P) as discussed in Section 3, Example II? Problem 6. What is the structure of CG(X)? Problem 7. Can the traditional parameters of convexity, Radon number, Caratheodory number, etc ., be related within the context of convex geometries? REFERENCES 1 . Aigner, M ., Combinatorial Theory, Springer-Verlag, New York, 1979 . 2 . Avann, S . P., Àpplications of Join-Irreducible Excess Functions to Semi-modular Lattices', Math . Ann. 142 (1961), 345-354 . 3 . Baclawski, K., 'Cohen-Macaulay Ordered Sets', J. Algebra 63 (1980), 226-258. 4. Bjorner, A., 'Shellable and Cohen-Macaulay Partially Ordered Sets', Trans. Amer. Math. Soc . 260 (1980), 159-183 . 5 . Bjorner, A ., 'Homotopy Type of Posets and Lattice Complementation', J. Comb . Theory (A) 30 (1981), 90-100 . 6. Bjorner, A ., Òn Matroids, Groups and Exchange Languages' (preprint), Stockholm, 1983 . 7 . Bjorner, A ., Garsia, A ., and Stanley, R., Àn Introduction to Cohen-Macaulay Partially Ordered Sets', in Ordered Sets (I . Rival (ed .)), D . Reidel, Dordrecht, 1982 . 8 . Bland, R. G . and Las Vergnas, M., 'Orientability of Matroids', J. Comb. Theory (B), 24 (1978),94-123 .


269

9 . Cordovil, R ., Òriented Matroids of Rank 3 and Arrangements of Pseudolines', Ann. Discrete Math . 17 (1983), 219-223 . 10. Crapo, H . H ., `Selectors', Adv . in Math . 54 (1984), 233-277 .

11 . Danaraj, G. and Klee, V ., 'Shellings of Spheres and Polytopes', Duke Math . J . 41 (1974), 443-451 . 12 . Dushnik, B . and Miller, E., `Partially Ordered Sets', Amer. J . Math . 63 (1941), 600-610 .

13 . Dilworth, R . P ., `Lattices with Unique Irreducible Decompositions', Ann . Math . 41 (1940), 771-777 . 14. Edelman, P. H ., `The Lattice of Convex Sets of an Oriented Matroid', J. Comb. Theory (B) 33 (1982), 239-244. 15 . Edelman, P . H ., 'Meet-Distributive Lattices and the Anti-exchange Closure', Alg . Univ. 10

(1980), 290-299 . 16 . Edelman, P. H., 'Zeta Polynomials and the Mobius Function', Europ . J. Comb . 1 (1980), 335-340 . 17 . Farber, M . and Jamison, R ., `Convexity in Graphs and Hypergraphs', Report CORR

83-46, University of Waterloo, 1983 . 18 . Folkman, J. and Lawrence, J., Òriented Matroids', J. Comb. Theory (B) 25 (1978), 199236 . 19 . Greene, C . and Kleitman, D . J ., `The Structure of Sperner k-Families', J. Comb. Theory

(A) 20 (1976),41-68 . 20. Goodman, J. E. and Pollack, R ., 'Helly-Type Theorems for Pseudoline Arrangements in P2', j . Comb . Theory (A) 32 (1982), 1-19 .

21 . Goodman, J . E . and Pollack, R., Òn the Combinatorial Classification of Nondegenerate Configurations in the Plane', J. Comb . Theory (A) 29 (1980), 120-235 . 22. Goodman, J . E . and Pollack, R ., À Theorem of Ordered Duality', Geom . Dedicata 12 (1982),63-74. 23 . Graham, R . L ., Simonovitz, M . and Sos, V. T., À Note on the Intersection Properties of Subsets of the Integers', J. Comb . Theory (A) 28 (1980), 107-110. 24 . Harary, F ., Graph Theory, Addison-Wesley, London, 1969 . 25 . Hoffman, A ., `Binding Constraints and Helly Numbers', Ann . N.Y. Acad . Sci., Second Int. Conf. Comb . Math . 319 (1979), 284-288 . 26 . Howorka, E., À Characterization of Ptolemaic Graphs ; Survey of Results', Proc . 8th SE Conf. Comb ., Graph Theory and Comp ., pp. 355-361 . 27 . Jamison-Waldner, R. E ., `Partition Numbers for Trees and Ordered Sets', Pacific J . Math. 96 (1981),115-140 . 28 . Jamison-Waldner, R . E., `Convexity and Block Graphs', Congressus Numerantium 33 (1981),129-142 . 29 . Jamison-Waldner, R . E., 'Tietze's Convexity Theorem for Semilattices and Lattices', Semigroup Forum 15 (1978), 357-373 . 30 . Jamison-Waldner, R . E ., À Perspective on Abstract Convexity : Classifying Alignments by Varieties', in Convexity and Related Combinatorial Geometry (D. C . Kay and M . Breen (eds)), Marcel Dekker, Inc ., New York, 1982 . 31 . Jamison-Waldner, R . E., À Convexity Characterization of Ordered Sets', Proc. 10th SE Conf Comb ., Graph Theory, and Comp . pp. 529-540. 32 . Jamison-Waldner, R . E., 'Copoints in Antimatroids', Proc . 11th SE Conf. Comb., Graph

Theory, and Comput ., Congressus Numerantium 29 (1980), 535-544. 33 . Korte, B . and Lovasz, L ., `Mathematical Structures Underlying Greedy Algorithms', in Fundamentals of Computation Theory, Lecture Notes in Computer Science 117, Springer, Berlin, 1981, pp . 205-209 . 34 . Korte, B . and Lovasz, L ., 'Greedoids - a Structural Framework for the Greedy Algorithms', Proc . Silver Jubilee Conf. on Combinatorics, Waterloo, June 1982, Academic Press, London .

270


35 . Korte, B . and Lovasz, L ., `Structural Properties of Greedoids', Combinatorica 3 (1983), 359-374 . 36. Korte, B. and Lovasz, L ., `Shelling Structures, Convexity and a Happy End', in Graph Theory and Combinatorics, Proceedings of the Cambridge Combinatorial Conference in honor of Paul Erdos, B . Bollobas (ed .), Academic Press, London (1984), 217-232 . 37 . Korte, B . and Lovasz, L ., `Basis Graphs of Greedoids and Two-Connectivity', Report No . 84324-OR, Institute of Operations Research, University of Bonn, 1984 . 38 . Las Vergnas, M ., `Convexity in Oriented Matroids', J. Comb . Theory (B) 29 (1980), 231-243 . 39. Rota, G.-C., Òn the Foundations of Combinatorial Theory : I . Theory of Mobius Functions', Z . Warsch . Verw . Gebiet 2 (1964), 340-368. 40. Saks, M . E . (personal communications), 1984 . 41 . Stanley, R. P ., `Combinatorial Reciprocity Theorems', Adv. in Math . 14 (1974), 194-253 . 42. Stanley, R . P ., `Balanced Cohen-Macaulay Complexes', Trans. Amer. Math. Soc. 249 (1979), 139-157 . 43 . Edelman, P . H . and Klingsberg, P ., The subposet lattice and the order polynomial, Europ. J. Comb ., 3 (1982), 341-346. 44. Berge, C ., Principles of Combinatorics, Academic Press, New York, 1970 .

Authors' addresses : Paul H . Edelman, Department of Mathematics, University of Pennsylvania, Philadelphia, P .A . 19104, U.S .A.

Robert E . Jamison, Department of Mathematical Sciences, Clemson University, Clemson, SC 29631, U.S .A . (Received, February 8, 1985)