The order versions of the Hahn--Banach Theorem and envelopes. I. Homogeneous functions

TL;DR

This paper explores the existence and construction of upper and lower envelopes for homogeneous functions within ordered sets, focusing on an order-algebraic framework without topology considerations.

Contribution

It provides a general formulation for envelopes of functions in ordered sets, specifically addressing homogeneous functions in an order-algebraic context.

Findings

Formulated conditions for the existence of envelopes.

Constructed methods for envelopes of homogeneous functions.

Focused on order-algebraic, topology-free setting.

Abstract

We present here the General formulation of the problem of existence and construction of upper and lower envelope for an arbitrary function with values from the completion of the ordered set ${\rm S}$ for a certain class of functions with values in ${\rm S}$. The task is parsed only for the simplest case of model class of homogeneous functions. Consider only order-algebraic version without the involvement of the topology.