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Extended-valued topical and anti-topical functions on semimodules

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.