Support theorems in abstract settings

TL;DR

This paper develops a broad framework for support theorems applicable to generalized convex functions between algebraic and ordered structures, introducing new concepts and conditions for completeness and convexity.

Contribution

It introduces new algebraic concepts like convex sets, extreme sets, and interior points in complex structures, extending support theorems to these generalized settings.

Findings

Established a general framework for support theorems in abstract algebraic settings.

Introduced new concepts of convexity and interior points in algebraic structures.

Derived support and extension theorems in classical and important settings.

Abstract

In this paper we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain space, we allow algebraic structures equipped with families of algebraic operations whose operations are mutually distributive with respect to each other. We introduce several new concepts in such algebraic structures, the notions of convex set, extreme set, and interior point with respect to a given family of operations, furthermore, we describe their most basic and required properties. In the context of the range space, we introduce the notion of completeness of a partially ordered set with respect to the existence of the infimum of lower bounded chains, we also offer several sufficient condition which imply this property. For instance, the order generated by a sharp cone in a vector space turns out to possess this completeness property. By taking several particular cases, we deduce support and extension theorems in various classical and important settings.