arXiv subjects
Publications and source records attributed to Artem N. Shevlyakov.
In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.
We describe all finite subsemigroups of a free left regular band of infinite rank. Moreover, we show applications of this result in algebraic geometry and model theory.
Equations over linearly ordered semilattices are studied. For any equation $t(X)=s(X)$ we find irreducible components of its solution set and compute the average number of irreducible components of all equations in $n$ variables.
We discuss new problems in universal algebraic geometry and explain them by boolean equations.
In the current paper we study extremal semilattices with respect to their equational properties. In the class $\mathbf{S}_n$ of all semilattices of order $n$ we find semilattices which have maximal (minimal) number of consistent equations. Moreover, we find a semilattice in $\mathbf{S}_n$ with maximal sum of numbers of solutions of equations.
A semigroup $S$ is called an equational domain (e.d.) if any finite union of algebraic sets over $S$ is algebraic. For a semigroup $S$ with a finite ideal we find the necessary and sufficient conditions to be an e.d.
A semigroup $S$ is called an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove some necessary and sufficient conditions for a completely simple semigroup to be an equational domain.
A semigroup $S$ is called an equational domain if any finite union of algebraic sets over $S$ is algebraic. For a finite simple semigroup we find necessary and sufficient conditions to be an equational domain. Moreover, we study semigroups with nontrivial center and prove that any such semigroup is not an equational domain.
We deal with equations over free semilattice of infinite rank and prove that any infinite consistent system of equations is equivalent to its finite subsystem. Moreover, we describe irreducible algebraic sets and solve some algorithmic problems.
A semigroup $S$ is an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove that every nontrivial semigroup in the standard language $\{\cdot\}$ is not an equational domain.
A semigroup $S$ is an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove that if an inverse semigroup $S$ is an equational domain in the extended language $\{\cdot,{}^{-1}\}\cup\{s|s\in S\}$ then $S$ is a group.
A semigroup $S$ is called an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove if a completely regular semigroup $S$ is an equational domain then $S$ is completely simple.
We study equations over boolean algebras with distinguished elements. We prove the criteria, when a boolean algebra is equationally Noetherian, weakly equationally Noetherian, $\mathbf{q}_ω$-compact or $\mathbf{u}_ω$-compact. Also we solve the problem of geometric equivalence in the class of boolean algebras with distinguished elements.