Extended-valued topical and anti-topical functions on semimodules

In the papers [16] and [17] we have studied functions defined on a b-complete idempotent semimodule X over a b-complete idempotent semifield K=(K,⊕,⊗), with values in K, where K may (or may not) contain a greatest element supK, and the residuation x/y is not defined for x∈X and y=infX. In the present paper we assume that K has no greatest element, then adjoin to K an outside “greatest element” ⊤=supK and extend the operations ⊕ and ⊗ from K to K¯:=K∪{⊤}, so as to obtain a meaning also for x/infX, for any x∈X, and study functions with values in K¯. In fact we consider two different extensions of the product ⊗ from K to K¯, denoted by ⊗ and ⊗˙ respectively, and use them to give characterizations of topical (i.e. increasing homogeneous, defined with the aid of ⊗) and anti-topical (i.e. decreasing anti-homogeneous, defined with the aid of ⊗˙) functions in terms of some inequalities. Next we introduce and study for functions f:X→K¯ their conjugates and biconjugates of Fenchel-Moreau type with respect to the coupling functions φ(x,y)=x/y, ∀x,y∈X, and ψ(x,(y,d)):=inf{x/y,d}, ∀x,y∈X, ∀d∈K¯, and use them to obtain characterizations of topical and anti-topical functions. In the subsequent sections we consider for the coupling functions φ and ψ some concepts that have been studied in Rubinov and Singer (2001) [11] and Singer (2004) [15] for the so-called “additive min-type coupling functions” πμ:Rmaxn×Rmaxn→Rmax and πμ:An×An→A respectively, where A is a conditionally complete lattice ordered group and πμ(x,y):=inf1⩽i⩽n(xi+yi), ∀x,y∈Rmaxn (or An). Thus, we study the polars of a set G⊆X for the coupling functions φ and ψ, and we consider for a function f:X→K¯ the notion of support set of f with respect to the set T˜ of all “elementary topical functions” t˜y(x):=x/y, ∀x∈X, ∀y∈X\{infX} and two concepts of support set of f at a point x0∈X. The main differences between the properties of the conjugations with respect to the coupling functions φ,ψ and πμ and between the properties of the polars of a set G with respect to the coupling functions φ,ψ and πμ are caused by the fact that while πμ is symmetric, with values only in Rmax (resp. A), φ and ψ are not symmetric and take values also outside Rmax (resp. A).