Розклади операцій медіальних і абелевих алгебр.

Abstract

Let A be an m x n matrix of variables. An n-ary operation f and m-ary operation g are said to satisfy the medial law if two results are the same: 1) an application of f to the rows of A then an application of g to the obtained column and 2) an application of g to the columns of A then an application of f to the obtained row. A universal algebra (A; Ω) is called: medial if every two operations from Ω satisfy the medial law; abelian if it is medial and has a one-element subalgebra. Criteria for being medial and for being Abelian are found for universal algebras (A; Ω) which have 0 Q and f Ω such that the term f(x 0 ,…, x n ) defines a quasigroup operation if all variables are 0 except x i and x p and it defines a permutation of Q if all variables are f(0,…,0) except x i or except x p for some different i, p.