An algebraic closure for barycentric algebras and convex sets

Abstract

Let A be an algebra (of an arbitrary finitary type), and let γ be a binary term. A pair (a, b) of elements of A will be called a γ-eligible pair if for each x in the subalgebra generated by {a, b} such that x is distinct from a there exists an element y in A such that b =  xyγ. We say that A is a γ-closed algebra if for each γ-eligible pair (a, b) there is an element c with b =  acγ. We call A a closed algebra if it is γ-closed for all binary terms γ that do not induce a projection.

Let T be a unital subring of the field of real numbers. Equipped with all the binary operations ${(x, y) \mapsto (1- p)x+py}$ for ${p \in T}$ and 0 <  p < 1, T becomes a mode, that is, an idempotent algebra in which any two term functions commute. In fact, the mode T is a (generalized) barycentric algebra. Let ${\mathcal{Q}(T)}$ denote the quasivariety generated by this mode.

Our main theorem asserts that each mode of ${\mathcal{Q}(T)}$ extends to a minimal closed cancellative mode, which is unique in a reasonable sense. In fact, we prove a slightly stronger statement. As corollaries, we obtain a purely algebraic description of the usual topological closure of convex sets, and we exemplify how to use the main theorem to show that certain open convex sets are not isomorphic.