Convexity and a Stone-type theorem for convex sets in abelian semigroup setting

TL;DR

This paper introduces two notions of convexity in abelian semigroups, explores their algebraic and set-theoretic properties, derives a convex hull formula, and proves a Stone-type separation theorem for disjoint convex sets.

Contribution

It presents new definitions of convexity in abelian semigroups and establishes a Stone-type separation theorem, extending convex analysis beyond traditional settings.

Findings

Two notions of convexity are introduced in abelian semigroups.

A formula for the convex hull in this setting is derived.

A Stone-type separation theorem for disjoint convex sets is proved.

Abstract

In this paper, two parallel notions of convexity of sets are introduced in the abelian semigroup setting. The connection of these notions to algebraic and to set-theoretic operations is investigated. A formula for the computation of the convex hull is derived. Finally, a Stone-type separation theorem for disjoint convex sets is established.