Semimodular functions and combinatorial geometries

A point-lattice £ being given, to any normalized, nondecreasing, integer-valued, semimodular function / defined on £, we can associate a class of combinatorial geometries called expansions of /. The family of expansions of / is shown to have a largest element for the weak map order, E(f), the free expansion of /. Expansions generalize and clarify the relationship between two known constructions, one defined by R. P. Dilworth, the other by J. Edmonds and G.-C. Rota. Further applications are developed for solving two extremal problems of semimodular functions: characterizing (1) extremal rays of the convex cone of real-valued, nondecreasing, semimodular functions defined on a finite set; (2) combinatorial geometries which are extremal for the decomposition into a sum. Introduction. S being a finite set, a real-valued function / defined on all subsets of S is said to be semimodular if and only if f(A)+f(B)>f(AuB)+f(AnB), \/A,BcS. The set of all semimodular, nondecreasing, real-valued functions defined on S forms a convex cone Gs. Such functions have occurred in a gametheoretical framework [18], and in [4] they are called alternating capacities of order 2. Their importance is especially remarkable in the context of combinatorial geometries, a theorem by Edmonds and Rota [12] stating that each integer-valued, semimodular nondecreasing function f on S defines a unique pregeometry (matroid) G(f, S) called the geometrization off. This theorem gives the motivation to investigate further connections between semimodular functions and combinatorial geometries, with the objective of developing new constructions of geometries. The major purpose of this work is to introduce and study a new class of geometric constructions on semimodular nondecreasing, integer-valued, normalized functions. Given any such function / defined on subsets of S, we Received by the editors October 21, 1975 and, in revised form, September 27, 1976. AMS (MOS) subject classifications (1970). Primary 05B35, 52A40; Secondary 06A20, 15A39. 1 This paper contains results from the author's Ph.D. dissertation (Massachusetts Institute of Technology, May 1975) written under the direction of Professor Gian-Carlo Rota. The author wishes to thank Professor Rota for his valuable help and also Professor Thomas Brylawski for many enlightening discussions. © American Mathematical Society 1978 355 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 356 HIEN QUANG NGUYEN consider the disjoint union X of the sets Xa, where for each a G S, \Xa\ = f(a). With the notation that for any subset A of S, XA = U aÊi4^i one °ftne main results (Theorem 1.3.3) of this paper is that % = {K\K c X,VA c 5, j AT n A^| > /04)} is the family of dependent sets of a pregeometry on X. This theorem guarantees the existence of a class of pregeometries associated to any such function /that we call expansions off. In §1, the operation of expansion is defined and some general results are derived. The main property is that for any function /of the cone 6S which is integer-valued and normalized, the operation of expanding / is always possible, and among all expansions of /, ordered by the weak map order, there is a largest element, the free expansion of /, E(f). Several characterizations of E(f) are derived, in terms of its circuits, bases and rank-function. An immediate important application of E(f) is to derive the operation of geometrization as a subgeometry: Edmonds and Rota's theorem becomes a consequence of the existence of the free expansion. Properties of geometrization are then studied and special consideration is given to the set S (G) of elements of the cone Gs which are integer-valued, nonnegative and whose geometrization defines a given pregeometry G. S(G) is characterized and the cases when S (G) is a finite set, and more particularly when S (G) has exactly two elements, are studied. §2 develops some applications of the idea of expansions to two extremal problems of the following nature. Given a certain decomposition D of elements of the set Qs, the question is to characterize the elements of Qs which are D-extremal or .D-irreducible, i.e. which cannot be decomposed into simpler elements of Qs. The first case we consider is the convex decomposition, i.e. an element / G Qs is reducible if and only if 3a„ a2 E R and 3/„/2 G &s, nonproportional to/such that/= zz,/, + a2f2. The problem is equivalent to characterizing the extremal rays of the cone Gs. After proving that a rank-function of a pregeometry is extremal if and only if the pregeometry is connected, we obtain a characterization of extremal integervalued, normalized elements of &s in terms of expansions (Theorem 2.1.9): such a function is not extremal if and only if some integer multiple of /has a proper disconnected expansion. The second case is a decomposition based on Edmonds' and Rota's theorem, called the sum-decomposition. In this case also the solution is obtained using properties of expansions (Theorem 2.2.1): a pregeometry G(r, s) is sum decomposable if and only if $>+(G) = {/|/ G S (G),f > 0} contains an element which has a disconnected expansion. 1. Expansion and geometrization. 1.1. Basic concepts and notations. This section presents a survey of the basic notions of combinatorial geometries needed in our work. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use SEMIMODULAR functions 357 A combinatorialpregeometry G(S),_or simply a pregeometry G, is a_set S together with a closure relation A -» A G (or A if no ambiguity) for A, A c S, which satisfies the following two axioms: Exchange Axiom. If a, b E S, A c S, and a EA u è-^i, then b EA U a. _ Finite Basis Property. If A c S, there is a finite subset A0cA such that A pregeometry is a geometry if 0 and all single-element subsets are closed. The/to of G (S) are the closed subsets of S. The set of all flats of G(S) ordered by inclusion is a geometric lattice, i.e. a semimodular point lattice with finite rank. A subset A c S is independent if for no a G A, A cA — a. If A is not independent, then A is dependent. If B c A c S and .4 c 7, we say that 7 spans A. A basis of ^4, for A c 5, is an independent subset of A which spans A. All bases of A have the same cardinality, r(A), the ran/: of A. If /4 is finite, the nullity of ^ is n(A) = \A\ r(A). The flats of G(S) of rank 1, 2, r(G) 1, /•(G) — 2 are called point, line, copoint, coline, respectively. A circuit is a minimal dependent set. A cyclic flat is a flat which is a union of circuits. We will use the following cryptomorphic definitions of a pregeometry G(S): Axioms of independent sets. A collection 5 of subsets of a set S forms the independent sets of a pregeometry G(S) if and only if: (1)1 E 5 and/ c/=>7 G 5; (2) A cS=» all maximal subsets of A which are elements of 5 have the same cardinality. Axioms of bases. A collection © of subsets of a set S is the set of bases of a pregeometry G (S) if and only if: (1) 3« G N, n > 0, such that V7 G ©, |7| = n; (2) if Bx, B2 G © and x G 7„ then By G 72 such that Bx x u y G © (basis exchange property). Axioms of the rank-function. An integer-valued function / defined on the subsets of a set S is the rank-function of a pregeometry on S if and only if: (l)/(0) = 0,/nondecreasing; (2)\/xES,0< f({x))< 1; (3) MA, Be S,f(A u 7) + /(^i n 7) < f(A) + f(B). A separator of G is a subset A c S such that r(S) = r(A) + r(S — A): every circuit of G is contained either in A or S — A and conversely. If G has no separators other than 0 and S, then G is connected. G is connected if and only if for any two points x,y of S, there is a circuit of G containing both. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 358 HIEN QUANG NGUYEN A c S, the subgeometry of G defined on A, G — A, is the pregeometry on A whose closure relation is U c A —> U n A. The contraction of G by A, G/A, is the pregeometry on S — A, with closure U c S — A-+U \J A— A. A point x G S is an isthmus of G if r(C7 — x) = r(G) — 1. x is a /oop of G if r(G/x) = r(G). Given two geometric lattices L, and L2, a sr/wzg map from L, to L2 is a function o: LX-*L2 which is supremum-preserving and cover-preserving. A strong map a between two pregeometries G (S) and H(T) is a strong map between the corresponding geometric lattices of flats of G(S) and H(T). With the expedient of adjoining a point 0 to each point set S and F, a determines a function b~ from the point set S U 0 to the point set F U 0, with 5(0) = 0. ä is said to extend to the strong map o. A function 5 from S U 0 to F u 0 such that 5(0) = 0 extends to a strong map from G (S) to H(T) if and only if the inverse image of any flat of H(T) is a flat of G(S). A function 5 from S u 0 to F u 0 extends to a weak map o between G (S) and H(T) if and only if for any set A c S such that 5(A) is independent and ö is one-one on A, then the subset y4 is independent. The set of all pregeometries defined on a set S is ordered by the weak map order "Gx < G2 if and only if the identity function is a weak map from G2 to G". Notations. We will use the following notation throughout the paper. S is a finite set. G(r, S) is a pregeometry on S with rank-function r. B (S) is the Boolean algebra of subsets of S. TBK(S) is the truncated Boolean algebra of rank k on S. £ stands for a lattice whose least element is denoted by 0 and largest element by 1. The order defining £ is denoted < while/I •< B for A, B G £ means that B covers A. The operations inf and sup are denoted A and VA subsemilattice £' of £ is a subset of £ which is closed under the operation /\:VA,B E t',A /\B G £'. Ce is the set of all real-valued, semimodular, nondecreasing functions defined on £. G% is the subset of Gt of normalized functions (/ G Gñ is normalized **/(0) = 0). lGt is the subset of (2e of integer-valued functions. IG% is the subset of 6g of integer-valued functions