Description of Super Associative Algebras With n-Quasigroup Operations

Janez Ušan

Let $\Sigma$ be a set operation over $Q$. Let also $w_{1} = w_{2}$ be a law in a description of which variables $x_{1},\dots,x_{s}$ are included, and also operational symbols $X_{1},\dots,X_{k}$, whose set of lengths is a subset of the set of lengths of operations from $\Sigma$. Then $(Q, \Sigma$) is said to be an algebra with the super identity $w_{1} = w_{2}$ if for every substitution of the variables $x_{1},\dots,x_{s}$ with elements of $Q$ and for every substitution of operational symbols $X_{1},\dots,X_{k} with operations from $\Sigma$ [with the corresponding lengths] $w_{1} = w_{2}$ becomes an equality in $(Q, \Sigma)$; [2]. Quasigroup algebras with associative superlaws were described by V.D. Belousov [5] (See also [16]). 3-quasigroup algebras with associative superlaws were primary described by Yu. M. Movsisyan ([9], p. 152-158). (Associative superlaws of hyperidentities of associativity; see also [15]). In the present paper, for n-quasigroup algebras with associative superlaws, the author was free to use the name: super associative algebras of n-quasigroup operations [briefly: $SAAnQ$]. In the paper, primary, in a unique way are described nontrivial $SAAnQ$ [briefly: $NetSAAnQ$] for every $nı N\backslash\{1\}$ with an exception of a case for $n = 2$. The crucial role in the mentioned description of $NetSAAnQ$ play the $\{1, n\}$-neutral and the inversing operations in an n-group. Starting with the mentioned description of $NetSAAnQ$, these algebras for $n\geq 3$ are finally described in terms of Hosszú-Gluskin algebras of order $n$.