Topical functions on semimodules and generalizations

We give first some characterizations of strongly supertopical respectively topical (that is, increasing strongly superhomogeneous, respectively increasing homogeneous) functions on a b-complete semimodule X over a b-complete idempotent semiring (respectively semifield) K=(K,⊕,⊗), with values in K, that improve and complement the main result of [12], . For example, we show that if K is a semifield and ε and e denote the neutral elements of K for ⊕ and ⊗ respectively, then every strongly supertopical function f:X→K satisfying f(infX)=ε is topical and that if (and only if) K≠{ε,e}, then every strongly supertopical function f:X→K is topical. We also give characterizations of strongly topical and topical functions with the aid of some inequalities. Next, generalizing [10], , we introduce elementary affine functions f:X→K and we apply them to obtain characterizations and a representation of topical functions. As a consequence, we obtain some characterizations of downward sets in X with the aid of elementary affine functions. Next we extend a result on topical functions f:Rn→R=(R,max,+) given in [12], to functions f:RI→R, where I is an arbitrary index set. Finally, we give characterizations of subtopical (i.e. increasing subhomogeneous) functions f:X→K, encompassing results of [10,11]. Our main tool is residuation theory.