POLYADIC OPERATION OF SPECIAL TYPE

Abstract

In the article the authors continue to study the polyadic operation η_{s, σ, k} that was earlier defined at the Cartesian power A^k of the nth groupoid < A, η > by the substitution of σ ∈ Sk and the nth operation η. The special case of the polyadic operation ηs, σ, k is the lth operation [ ]_{l, σ, k} that is defined by one of the authors for any integer k ≥ 2, l ≥ 2 and for any substitution of the σ set {1, …, k} at the Cartesian power A^k of the semigroup A. In turn, the special case of the lth operation [ ]_{l, σ, k} consists of two polyadic operations by E. Post, one of which he defined at the Cartesian power of the symmetric group and the second – at the Cartesian power of the general linear group over the field of complex numbers. The properties of the operations η_{s, σ, k} are studied in the article. In particular, a new proof of the associativity of the polyadic operation η_{s, σ, k} was obtained.