Extension theorems for functional equations with bisymmetric operations

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9 Citations (Scopus)

Abstract

In this paper we deal with the extension of the following functional equation f(x) = M(f(m1(x,y)),...,f(mk(x,y))) (x,y ∈ K), (*) where M is a k-variable operation on the image space Y, m 1,...,mk are binary operations on X, K ⊂ X is closed under the operations m1,...,mk, and f : K → Y is considered as an unknown function. The main result of this paper states that if the operations m1,...,mk, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m1,...,mk)-affine hull K* of K, such that (*) holds over K*. (The set K* is defined as the smallest subset of X that contains K and is (m1,..., mk)-afrine, i.e., if x ∈ X, and there exists y ∈ K* such that m1(x, y),...,m k(x, y) ∈ K* then x ∈ K*). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.

Original languageEnglish
Pages (from-to)266-291
Number of pages26
JournalAequationes Mathematicae
Volume63
Issue number3
DOIs
Publication statusPublished - Jan 1 2002

Keywords

  • Bisymmetric operation
  • Extension theorem
  • Functional equation

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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