On the Pexider equation
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.
scientific article; zbMATH DE number 3899468