Discrete Applied Mathematics 147 (2005) 69–79
Closure spaces that are not uniquely generated
Robert E. Jamison
, John L. Pfaltz
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA
Department of Computer Science, University of Virginia, Charlottesville, VA 22903, USA
Received 30 August 2000; received in revised form 20 July 2002; accepted 21 June 2004
Because antimatroid closure spaces satisfy the anti-exchange axiom, it is easy to showthat they are
uniquely generated. That is, the minimal set of elements determining a closed set is unique. A prime
example is a discrete convex geometry in Euclidean space where closed sets are uniquely generated
by their extreme points. But, many of the geometries arising in computer science, e.g. the world wide
web or rectilinear VLSI layouts are not uniquely generated. Nevertheless, these closure spaces still
illustrate a number of fundamental antimatroid properties which we demonstrate in this paper. In
particular, we examine both a pseudo-convexity operator and the Galois closure of formal concept
analysis.In the latter case, weshowhowthese principles can beused to automaticallyconverta formal
concept lattice into a system of implications.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Galois closure; Antimatroid; Convex; Concept lattice; Disjunctive implication
Matroids and antimatroids can be studied either in terms of a family
of feasible sets
and a shelling operator
[1,11], or in terms of a collection
of closed sets and a closure
[3,14]. There exists a considerable amount of confusion, and an equally great
richness, because these are two distinct approaches to precisely the same concepts. Given
an antimatroid universe, U, every feasible set F ∈
is the complement of a closed set
E-mail address: email@example.com (R.E. Jamison)
Supported in part by DOE Grant DE-FG05-95ER25254.
0166-218X/$-see front matter © 2004 Elsevier B.V.All rights reserved.