×

The theory of convex geometries. (English) Zbl 0577.52001

An alignment L (also called a domain finite convexity structure) on a nonempty set X is the family of subsets of X, called convex sets, closed under an algebraic closure operator L. In this paper, X is always finite and L also is antiexchange, i.e., if K is convex, p,q\(\not\in K\), \(p\neq q\), then \(q\not\in L(K\cup p)\) implies \(p\not\in L(K\cup q)\). Such a pair (X,L) is called a convex geometry.
In this well-presented survey the authors develop the foundations and give an up-to-date account of these geometries. Several characterizations are presented. The standard examples of convexity structures on finite sets are shown to be convex geometries. Also some new examples are offered. The lattice of convex subsets of a convex geometry is meet- distributive and this characterizes convex geometries. Various additional properties of these lattices are discussed. Operations on convex geometries, viz., joins and minors, are considered. The paper concludes with a list of open questions concerning convex geometries.
Reviewer: J.H.M.Whitfield

MSC:

52A01 Axiomatic and generalized convexity
PDFBibTeX XMLCite
Full Text: DOI