Вадим Усольцев - Academia.edu (original) (raw)
Вадим Усольцев
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Papers by Вадим Усольцев
Chebyshevskii sbornik, 2021
In this paper we introduce the concept of Rees closure for subalgebras of universal algebras. We ... more In this paper we introduce the concept of Rees closure for subalgebras of universal algebras. We denote by △𝐴 the identity relation on 𝐴. A subalgebra 𝐵 of algebra 𝐴 is called a Rees subalgebra whenever 𝐵2 ∪ △𝐴 is a congruence on 𝐴. A congruence 𝜃 of algebra 𝐴 is called a Rees congruence if 𝜃 = 𝐵2 ∪△𝐴 for some subalgebra 𝐵 of 𝐴. We define a Rees closure operator by mapping arbitrary subalgebra 𝐵 of algebra 𝐴 into the smallest Rees subalgebra that contains 𝐵. It is shown that in the general case the Rees closure does not commute with the operation ∧ 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 𝐴 is called a Rees simple algebra if any Rees congruence on 𝐴 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.
Chebyshevskii sbornik, 2021
In that paper we study atoms of congruence lattices and subdirectly irreducibility of algebras wi... more In that paper we study atoms of congruence lattices and subdirectly irreducibility of algebras with one operator and the main symmetric operation. A ternary operation 𝑑(𝑥, 𝑦, 𝑧) satisfying identities 𝑑(𝑥, 𝑦, 𝑦) = 𝑑(𝑦, 𝑦, 𝑥) = 𝑑(𝑦, 𝑥, 𝑦) = 𝑥 is called a minority operation. The symmetric operation is a minority operation defined by specific way. An algebra 𝐴 is called subdirectly irreducible if 𝐴 has the smallest nonzero congruence. An algebra with operators is an universal algebra whose signature consists of two nonempty non-intersectional parts: the main one which can contain arbitrary operations, and the additional one consisting of operators. The operators are unary operations that act as endomorphisms with respect to the main operations, i.e., one that permutable with main operations. A lattice 𝐿 with zero is called atomic if any element of 𝐿 contains some atom. A lattice 𝐿 with zero is called atomistic if any nonzero element of 𝐿 is a join of some atom set. It shown that congrue...
Успехи математических наук, 2008
Chebyshevskii sbornik, 2021
In this paper we introduce the concept of Rees closure for subalgebras of universal algebras. We ... more In this paper we introduce the concept of Rees closure for subalgebras of universal algebras. We denote by △𝐴 the identity relation on 𝐴. A subalgebra 𝐵 of algebra 𝐴 is called a Rees subalgebra whenever 𝐵2 ∪ △𝐴 is a congruence on 𝐴. A congruence 𝜃 of algebra 𝐴 is called a Rees congruence if 𝜃 = 𝐵2 ∪△𝐴 for some subalgebra 𝐵 of 𝐴. We define a Rees closure operator by mapping arbitrary subalgebra 𝐵 of algebra 𝐴 into the smallest Rees subalgebra that contains 𝐵. It is shown that in the general case the Rees closure does not commute with the operation ∧ 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 𝐴 is called a Rees simple algebra if any Rees congruence on 𝐴 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.
Chebyshevskii sbornik, 2021
In that paper we study atoms of congruence lattices and subdirectly irreducibility of algebras wi... more In that paper we study atoms of congruence lattices and subdirectly irreducibility of algebras with one operator and the main symmetric operation. A ternary operation 𝑑(𝑥, 𝑦, 𝑧) satisfying identities 𝑑(𝑥, 𝑦, 𝑦) = 𝑑(𝑦, 𝑦, 𝑥) = 𝑑(𝑦, 𝑥, 𝑦) = 𝑥 is called a minority operation. The symmetric operation is a minority operation defined by specific way. An algebra 𝐴 is called subdirectly irreducible if 𝐴 has the smallest nonzero congruence. An algebra with operators is an universal algebra whose signature consists of two nonempty non-intersectional parts: the main one which can contain arbitrary operations, and the additional one consisting of operators. The operators are unary operations that act as endomorphisms with respect to the main operations, i.e., one that permutable with main operations. A lattice 𝐿 with zero is called atomic if any element of 𝐿 contains some atom. A lattice 𝐿 with zero is called atomistic if any nonzero element of 𝐿 is a join of some atom set. It shown that congrue...
Успехи математических наук, 2008