The arity of convex spaces

Abstract

The arity of convex spaces is a numerical feature which shows the ability of finite subsets spanning to the whole space via the hull operators. This paper gives it a formal and strict definition by introducing the truncation of convex spaces. The relations that between the arity of quotient spaces and the original spaces, that between the arity of subspaces and superspaces, that between the arity of product spaces and factors spaces, and that between the arity of disjoint sums and term spaces, are systematically studied. A mistake of a formula in [M. Van De Vel, Theory of Convex Structures, North-Holland, Amsterdam, 1993] is corrected. It is shown that a convex space is Alexandrov iff its arity is 1. The convex structures with arity ≤n are equivalent to structured sets with n-restricted hull operators.