Barycentric algebras and beyond

Abstract

Barycentric algebras are fundamental for modeling convex sets, semilattices, affine spaces and related structures. This paper is part of a series examining the concept of a barycentric algebra in detail. In preceding work, threshold barycentric algebras were introduced as part of an analysis of the axiomatization of convexity. In the current paper, the concept of a threshold barycentric algebra is extended to threshold affine spaces. To within equivalence, these algebras comprise barycentric algebras, commutative idempotent entropic magmas, and affine spaces, all defined over a subfield of the field of real numbers. Many properties of threshold barycentric algebras extend to threshold affine spaces.