On the Pexider equation
type: Item
instance of: scholarly article
title: On the Pexider equation
zbMATH Open document ID: 0564.39004
DOI: 10.1007/BF02189409
author: J. Blot
published in: Aequationes Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137056
review text: The Pexider equation \((1)\quad f(x*y)=g(x).h(y)\) corresponds to the notion of homotopy which has been used primarily in the theory of quasigroups. In this paper the general solution of equation (1) is found in the case where * is a mapping from \(S_ 1\times S_ 2\) to S and all the sets involved are nonempty \(((S,S_ 1,S_ 2,*)\) is called a generalized groupoid). Next it is shown that solutions of (1) involving homomorphisms of groupoids do not depend on the structures being equationally defined. Homomorphisms appear, in particular, in formulas for the general solution of (1) when f,g,h: \(S\to T\), and (T,.) is a group. The corresponding results are due to \textit{J. Aczél} [Publ. Inst. Math., Nouv. Ser. 4(18), 77-80 (1964; Zbl 0123.108)] (if (S,*) is an Abelian semigroup), \textit{E.Vincze} [Studia Univ. Babeş-Bolyai, Ser. I 7 (1962), No.1, 103-106 (1963; Zbl 0117.107)] (if (S,*) is a semigroup with the property \(a*S=S*b=S\) for some a,b\(\in S)\) and \textit{M. A. Taylor} [Acta Math. Acad. Sci. Hung. 36, 211-213 (1980; Zbl 0484.39004)] (if (S,*) is an arbitrary semigroup). A number of results for equation (1) on semigroups of special kinds is also given.
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 20L05
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 3899468
zbMATH Keywords: Pexider equation
zbMATH Keywords: quasigroups
zbMATH Keywords: general solution
zbMATH Keywords: groupoid
reviewed by: Bogdan A. Choczewski
MaRDI profile type: Publication
cites work: Q5514010
cites work: Q5735494
cites work: Q5637102
cites work: Q5515384
cites work: Q4769064
cites work: Q5665375
cites work: Q5552183
cites work: A pexider equation for functions defined on a semigroup
cites work: Heterogeneous algebras
OpenAlex ID: W2022399791
description: scientific article; zbMATH DE number 3899468