Rees algebras and Rees congruence algebras of one class of algebras with operator and basic near-unanimity operation

V. L. Usol'tsev

Čebyševskij sbornik, Tome 17 (2016) no. 4, pp. 157-166

Résumé

The concept of Rees congruence was originally introduced for semigroups. R. Tichy generalized this concept to universal algebras. Let $A$ be an universal algebra. Denote by $\bigtriangleup$ the identity relation on $A$. Any congruence of the form $B^2 \cup \bigtriangleup$ on $A$ for some subalgebra $B$ of $A$ is called a Rees congruence. Subalgebra $B$ of $A$ is called a Rees subalgebra whenever $B^2 \cup \bigtriangleup$ is a congruence on $A$. An algebra $A$ is called a Rees algebra if its every subalgebra is a Rees one. In this paper we introduce concepts of Rees simple algebra and Rees congruence algebra. A non-one-element universal algebra $A$ is called Rees simple algebra if any Rees congruence on $A$ is trivial. An universal algebra $A$ is called Rees congruence algebra if any congruence on $A$ is Rees congruence. Universal algebra is called an algebra with operators if it has an additional set of unary operations acting as endomorphisms with respect to basic operations. For algebras with one operator and an arbitrary basic signature some conditions to be Rees algebra are obtained. Necessary condition under which algebra of the same class is Rees congruence algebra is given. For algebras with one operator and a connected unary reduct that has a loop element and does not contain the nodal elements, except, perhaps, the loop element necessary condition for their Rees simplicity are obtained. A n-ary operation $\varphi$ ($n \geqslant 3$) is called near-unanimity operation if it satisfies the identities $\varphi(x, \ldots, x, y) = \varphi(x, \ldots, x, y, x) = \ldots =$ $\varphi(y, x, \ldots, x)=x$. If $n=3$ then operation $\varphi$ is called a majority operation. Rees algebras and Rees congruence algebras of class algebras with one operator and basic near-unanimity operation $g^{(n)}$ which defined as follows $g^{(3)}(x_1,x_2,x_3)=m(x_1,x_2,x_3)$, $g^{(n)}(x_1,x_2, \ldots,x_n) = m(g^{(n-1)}(x_1,x_2, \ldots,x_{n-1}),x_{n-1},x_n)$ $(n>3)$ are fully described. Under $m(x_1,x_2,x_3)$ we mean here a majority operation which permutable with unary operation and which was defined by the author on arbitrary unar according to the approach offered by V. K. Kartashov. Bibliography: 16 titles.

Détail

Keywords: Rees algebra, Rees congruence, Rees simple algebra, Rees congruence algebra, algebra with operators, near-unanimity operation.