REES ALGEBRAS AND REES CONGRUENCE ALGEBRAS OF ONE CLASS OF ALGEBRAS WITH OPERATOR AND BASIC NEAR-UNANIMITY OPERATION
The concept of Rees congruence was originally introduced for semigroups. R. Tichy generalized this concept to universal algebras. Let be an universal algebra. Denote by △ the identity relation on 𝐴. Any congruence of the form 𝐵2 ∪△ on for some subalgebra of is called a Rees congruence. Subalgebra of is called a Rees subalgebra whenever 𝐵2 ∪△ is a congruence on 𝐴. An algebra 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 is called Rees simple algebra if any Rees congruence on is trivial. An universal algebra is called Rees congruence algebra if any congruence on 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 (𝑛 3) is called near-unanimity operation if it satisfies the identities 𝜙(𝑥,...,𝑥, 𝑦)= 𝜙(𝑥,...,𝑥, 𝑦,𝑥)=... = 𝜙(𝑦,𝑥,...,𝑥)= 𝑥. If =3 then operation 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)(𝑥1,𝑥2,𝑥3)= 𝑚(𝑥1,𝑥2,𝑥3), 𝑔(𝑛)(𝑥1,𝑥2,...,𝑥𝑛)= 𝑚(𝑔(𝑛-1)(𝑥1,𝑥2,...,𝑥𝑛-1),𝑥𝑛-1,𝑥𝑛) (𝑛 > 3) are fully described. Under 𝑚(𝑥1,𝑥2,𝑥3) 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.