Extended-valued topical and anti-topical functions on semimodules

TL;DR

This paper extends the theory of topical and anti-topical functions on semimodules by introducing an outside greatest element to study extended functions, their conjugates, and polars, enriching the mathematical framework for these functions.

Contribution

It introduces two extensions of the product to define and characterize extended topical and anti-topical functions, along with their conjugates and polars, in the context of semimodules over semifields.

Findings

Characterization of extended topical and anti-topical functions via inequalities

Development of conjugates and biconjugates of Fenchel-Moreau type

Analysis of polars and support sets in the extended framework

Abstract

In previous papers we have studied topical functions f:X-> K and related classes of functions, where X is a b-complete semimodule over an idempotent b-complete semifield K. Without essential restriction of the generality, we assume that K has no greatest element sup K, and hence for x in X and y=inf X the residuation x/y is not defined. Now we adjoin to K an outside "greatest element" top =sup K, and extending in a suitable way the operations of multiplication and addition from K to bar K, which is the union of K and top, we study "extended functions" f:X-> bar K. Actually, we give two different extensions of the product from K to bar K, so as to obtain a meaning also for the residuation x/inf X, with any x in X, and in particular for inf X/inf X, and use them to give characterizations of topical (i.e. increasing homogeneous, defined with the aid of the first product) and anti-topical (i.e. decreasing anti-homogeneous, defined with the aid of the second product) functions f:X-> bar K, by some inequalities. Next we study for functions f:X -> bar K their conjugates and biconjugates of Fenchel-Moreau type with respect to the coupling functions phi(x,y)=x/y x,y in X and psi(x,(y,d)):=inf {x/y,d}; x,y in X,d in bar K, and obtain characterizations of topical and anti-topical functions. In the subsequent sections we consider the polars of a subset G in X for the coupling functions phi and psi, and the support set of a function f:X->K with respect to the set tilde T of all "elementary topical functions" tilde t_{y}(x):=x/y, x in X, y in X-{inf X} and two concepts of support set of f:X-> bar K at a point x_0 in X.