On Rees closure in some classes of algebras with an operator
Π. Π. Π£ΡΠΎΠ»ΡΡΠ΅Π² Β· Π§Π΅Π±ΡΡΠ΅Π²ΡΠΊΠΈΠΉ ΡΠ±ΠΎΡΠ½ΠΈΠΊ Β· 2021
In this paper we introduce the concept of Rees closure for subalgebras of universal algebras. We denote by β³π΄ the identity relation on π΄. A subalgebra π΅ of algebra π΄ is called a Rees subalgebra whenever π΅2 βͺ β³π΄ is a congruence on π΄. A congruence π of algebra π΄ is called a Rees congruence if π = π΅2 βͺβ³π΄ for some subalgebra π΅ of π΄. We define a Rees closure operator by mapping arbitrary subalgebra π΅ of algebra π΄ into the smallest Rees subalgebra that contains π΅. It is shown that in the general case the Rees closure does not commute with the operation β§ on the lattice of subalgebras of universal algebra. Consequently, in the general case, a lattice of Rees subalgebras is not a sublattice of lattice of subalgebras. A non-one-element universal algebra π΄ is called a Rees simple algebra if any Rees congruence on π΄ is trivial. We characterize Rees simple algebras in terms of Rees closure. 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. We described Rees simple algebras in some subclasses of the class of algebras with one operator and a ternary basic operation. For algebras from these classes, the structure of lattice of Rees subalgebras is described. Necessary and sufficient conditions for the lattice of Rees subalgebras of algebras from these classes to be a chain are obtained.