### Abstract

In this paper we deal with the extension of the following functional equation f(x) = M(f(m1(x,y)),...,f(m_{k}(x,y))) (x,y ∈ K), (*) where M is a k-variable operation on the image space Y, m _{1},...,m_{k} are binary operations on X, K ⊂ X is closed under the operations m_{1},...,m_{k}, and f : K → Y is considered as an unknown function. The main result of this paper states that if the operations m1,...,m_{k}, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m1,...,m_{k})-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 (m_{1},..., m_{k})-afrine, i.e., if x ∈ X, and there exists y ∈ K* such that m_{1}(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 language | English |
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Pages (from-to) | 266-291 |

Number of pages | 26 |

Journal | Aequationes Mathematicae |

Volume | 63 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2002 |

### Keywords

- Bisymmetric operation
- Extension theorem
- Functional equation

### ASJC Scopus subject areas

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