Reducibility and uniform reducibility of algebraic operations

B. R. Frenkin

Sbornik. Mathematics, Tome 24 (1974) no. 3, pp. 373-384

Résumé

This paper is devoted to a study of the conditions under which one algebraic operation can be expressed in terms of others by some arrangement of parentheses. The terminology is mainly that of Frenkin (RZhMat., 1972, 2A235). It is shown that the class of $\sigma$-reducible $n$-groupoids is axiomatizable, but not elementary, and the class of $\tau$-uniformly reducible $n$-groupoids is not axiomatizable; a criterion for $\tau$-uniform reducibility in terms of pseudo-isotopies (a generalization of the concept of isotopy) between $\tau$-reducing operations is obtained. It is shown that a free $n$-groupoid of finite rank is not $\tau$-uniformly reducible, but one of infinite rank is $\tau$-uniformly reducible; as a consequence, any $n$-groupoid is a homomorphic image of one which is $\tau$-uniformly reducible. Some results on algebras with unary operations are also obtained. Bibliography: 7 titles.