Topical functions on semimodules and generalizations

Abstract

Elementary affine function Quasi-elementary affine function ∧-affine function Downward setWe 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 (inf X) = 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 : R n → R = (R, max, +) given in [12], to functions f : R I → 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.

Extended-valued topical and anti-topical functions on semimodules

“…In [17,Theorem 5] we have shown that if (X, K) is a pair satisfying (A0 ′ ), (A1), then a function f : X → K is topical if and only if f (inf X) = ε and…”

“…c) ε −1 and ⊤ −1 are called "inverses" only by abuse of language, as shown by (13) and (14). d) We shall see that with the above definition, the notions and results of [16,17] on functions f : X → K, where X is a semimodule over K, admit extensions in the above sense to functions f : X → K, for the extended product ⊗ of (10), (11). Therefore in the sequel whenever we shall refer to a result of [17] or [16], we shall understand, without any special mention, its extension (using the above conventions) to functions f : X → K, for the extended product ⊗ on K.…”

“…In the previous papers [16] and [17], attempting to contribute to the construction of a theory of functional analysis and convex analysis in semimodules over semifields, we have studied topical functions f : X → K and related classes of functions, where X is a b-complete idempotent semimodule over a b-complete idempotent semifield K. We recall that f : X → K is called topical if it is increasing (i.e., the relations x ′ , x ′′ ∈ X, x ′ ≤ x ′′ imply f (x ′ ) ≤ f (x ′′ ), where ≤ denotes the canonical order on K, respectively on X, defined by λ ≤ µ ⇔ λ ⊕ µ = µ, ∀λ ∈ K, ∀µ ∈ K, respectively by x ≤ y ⇔ x ⊕ y = y, ∀x ∈ X, ∀y ∈ X), and homogeneous (i.e., f (λx) = λf (x) for all x ∈ X, λ ∈ K, where λx := λ ⊗ x, λf (x) = λ ⊗ f (x); the fact that we use the same notations for addition ⊕ both in K and in X and for multiplication ⊗ both in K and in K × X will lead to no confusion). These definitions will be used also when K is replaced by R = ((−∞, +∞), ⊕ = max, ⊗ = +) although it is not a semiring, and X is replaced by R n .…”

“…Also, as has been observed in [11], many results on R n remain valid, essentially with the same proofs, for X = R I max , the set of all bounded vectors x = (x i ) i∈I where I is an arbitrary index set and x i ∈ R, ∀i ∈ I, sup i∈I |x i | < +∞, endowed with the componentwise semimodule operations x ′ ⊕x ′′ := (max(x ′ i , x ′′ i )) i∈I , λx = (λx i ) i∈I and the componentwise order relation x ′ ≤ x ′′ ⇔ x ′ i ≤ x ′′ i , ∀i ∈ I. One of the main tools in [16] and [17] has been residuation. We recall that by (A0 ′ ) and (A1), for each y ∈ X\{inf X} (hence such that the set {λ ∈ K|λy ≤ x} is (order-) bounded from above, by (A1)) there exists the residuation operation / defined by x/y := max{λ ∈ K|λy ≤ x}, ∀x ∈ X, ∀y ∈ X\{inf X},…”

Characterizations of minimal elements of topical functions on semimodules with applications

“…Recently, topical functions f : X −→ K and related classes of functions have studied in [4,13,18,19,21,22,23,25,26], where X is a b-complete idempotent semimodule over a b-complete idempotent semifield K. We recall that a function f : X −→ K is called topical if it is increasing (i.e., the relations x , x ∈ X, x ≤ x imply f (x ) ≤ f (x ), where ≤ denotes the canonical order on X, respectively on K, defined by x ≤ y if and only if x ⊕ y = y for all x ∈ X and all y ∈ X, respectively by λ ≤ μ if and only if λ ⊕ μ = μ for all λ ∈ K and all μ ∈ K), and homogeneous (i.e., f (λx) = λf (x) for all x ∈ X and all λ ∈ K, where λx := λ ⊗ x and λf (x) := λ ⊗ f (x); the fact that we use the same notations for addition ⊕ both in X and in K and for multiplication ⊗ both in K × X and in K will lead to no confusion).…”