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Mikhail V. Volkov

Publications and source records attributed to Mikhail V. Volkov.

At least 19 recordsLinked to original sources
arXiv2026

Completely Reachable Road Coloring

We characterize the digraphs that admit an edge labeling by letters from a finite alphabet such that the resulting labeled digraph is a completely reachable automaton. This class of digraphs can be recognized in polynomial time. In contrast, we show that, for every fixed alphabet size, the problem of deciding whether a digraph admits an edge labeling with the same property is NP-complete. We also classify the digraphs for which every edge labeling results in a completely reachable automaton.

cs.FL↗
arXiv2026

Adversarial Synchronization

We study a variant of the synchronization game on finite deterministic automata. In this game, Alice chooses one input letter of an automaton $A$ on each of her moves while Bob may respond with an arbitrary finite word over the input alphabet of $A$; Alice wins if the word obtained by interleaving her letters with Bob's responses resets $A$. We prove that if Alice has a winning strategy in this game on $A$, then $A$ admits a reset word whose length is strictly smaller than the number of states of $A$. In contrast, for any $k\ge 1$, we exhibit automata with shortest reset-word length quadratic in the number of states, on which Alice nevertheless wins a version of the game in which Bob's responses are restricted to arbitrary words of length at most $k$. We provide polynomial-time algorithms for deciding the winner in various synchronization games, and we analyze the relationships between variants of synchronization games on fixed-size automata.

cs.FL↗
arXiv2025

The finite basis problem for the endomorphism semirings of finite semilattices

For every semilattice $\mathcal{A}=(A,+)$, the set $\mathrm{End}(\mathcal{A})$ of its endomorphisms forms a semiring under pointwise addition and composition. We prove that that if $\mathcal{A}$ is finite, then the endomorphism semiring $\mathrm{End}(\mathcal{A})$ has a finite identity basis if and only if $|A|\le 2$.

math.RA↗
arXiv2025

Identities of triangular Boolean matrices

We give a combinatorial characterization of the identities holding in the semiring of all upper triangular Boolean $n\times n$-matrices and apply the characterization to computational complexity of identity checking, finite axiomatizability of equational theories, and algebraic descriptions of certain classes of recognizable languages.

math.GR↗
arXiv2025

Embedding lattices of quasivarieties of periodic groups into lattices of additively idempotent semiring varieties: An algebraic proof

A general result by Jackson (Flat algebras and the translation of universal Horn logic to equational logic, J. Symb. Log. 73(1) (2008) 90--128) implies that the lattice of all quasivarieties of groups of exponent dividing $n$ embeds into the lattice $L(\mathbf{Sr}_n)$ of all varieties of additively idempotent semirings whose multiplicative semigroups are unions of groups of exponent dividing $n$; the image of this embedding is an interval in $L(\mathbf{Sr}_n)$. We provide a new, direct, and purely algebraic proof of these facts and present a new identity basis for the top variety of the interval. In addition, we obtain new information about the lattice $L(\mathbf{Sr}_n)$, demonstrating that the properties of the lattice for $n\ge 3$ differ drastically from those previously known when $n=1$ or $2$.

math.GR↗
arXiv2024

The finite basis problem for endomorphism semirings of finite chains

For every semilattice $\mathcal{S}=(S,+)$, the set $\mathrm{End}(\mathcal{S})$ of its endomorphisms forms a semiring under point-wise addition and composition. We prove that the semiring of all endomorphisms of the 3-element chain has no finite identity basis. This, combined with earlier results by Dolinka (The finite basis problem for endomorphism semirings of finite semilattices with zero, Algebra Universalis 61, 441-448 (2009)), gives a complete solution to the finite basis problem for semirings of the form $\mathrm{End}(\mathcal{S})$ where $\mathcal{S}$ is a finite chain.

math.GR↗
arXiv2024

Don's conjecture for binary completely reachable automata: an approach and its limitations

A deterministic finite automaton in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word is called completely reachable. It was conjectured that in each completely reachable automaton with $n$ states, every set of $k>0$ states is the image of a word of length at most $n(n-k)$. We confirm the conjecture for completely reachable automata with two input letters satisfying certain restrictions on the action of the letters.

cs.FL↗
arXiv2024

Strongly nonfinitely based monoids

We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite semigroup.

math.GR↗
arXiv2023

The Burnside ai-semiring variety defined by $x^n\approx x$

Let ${\bf Sr}(n, 1)$ denote the ai-semiring variety defined by the identity $x^n\approx x$, where $n>1$. We characterize all subdirectly irreducible members of a semisimple subvariety of ${\bf Sr}(n, 1)$. Based on this result, we prove that ${\bf Sr}(n, 1)$ is hereditarily finitely based (resp., hereditarily finitely generated) if and only if $n<4$ and that the lattice of subvarieties of ${\bf Sr}(n, 1)$ is countable if and only if $n<4$. Also, we show that the class of all locally finite members of ${\bf Sr}(n, 1)$ forms a variety and so we affirmatively answer the restricted Burnside problem for ${\bf Sr}(n, 1)$. In addition, we provide a simplified proof of the main result obtained by Gajdoš and Kuřil (Semigroup Forum 80: 92--104, 2010).

math.GR↗
arXiv2023

Semiring and involution identities of powers of inverse semigroups

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis.

math.GR↗
arXiv2023

Catalan monoids inherently nonfinitely based relative to finite $\mathcal{R}$-trivial semigroups

We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite $\mathcal{R}$-trivial semigroup. This provides unified proofs for several known facts and leads to a bunch of new results on the Finite Basis Problem for finite $\mathcal{R}$- and $\mathcal{J}$-trivial semigroups.

math.GR↗
arXiv2022

Remark on the identities of the grammic monoid with three generators

Grammic monoids have recently been introduced by Christian Choffrut in terms of the action of the free monoid over a fixed ordered alphabet $X$ on the set of rows of Young tableaux filled with elements from $X$ via Schensted's insertion. For $X=\{a,b,c\}$ with $a<b<c$, Choffrut has identified the grammic monoid on $X$ with the quotient of the plactic monoid on $X$ over the congruence generated by the pair $(bacb,cbab)$. Since $cbab\ne bcab$ in the latter monoid, the quotient is proper. We show that, nevertheless, the plactic and the grammic monoids with three generators satisfy the same identities.

math.GR↗
arXiv2022

Binary completely reachable automata

We characterize complete deterministic finite automata with two input letters in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word. The characterization leads to a polynomial-time algorithm for recognizing this class of automata.

cs.FL↗
arXiv2022

Semiring and involution identities of power groups

For every group $G$, the set $\mathcal{P}(G)$ of its subsets forms a semiring under set-theoretical union $\cup$ and element-wise multiplication $\cdot$ and forms an involution semigroup under $\cdot$ and element-wise inversion ${}^{-1}$. We show that if the group $G$ is finite, non-Dedekind, and solvable, neither the semiring $(\mathcal{P}(G),\cup,\cdot)$ nor the involution semigroup $(\mathcal{P}(G),\cdot,{}^{-1})$ admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.

math.GR↗
arXiv2022

Identities of the stylic monoid

We observe that for each $n\ge 2$, the identities of the stylic monoid with $n$ generators coincide with the identities of $n$-generated monoids from other distinguished series of $\mathscr{J}$-trivial monoids studied in the literature, e.g., Catalan monoids and Kiselman monoids. This solves the Finite Basis Problem for stylic monoids.

math.GR↗
arXiv2022

Semiring identities of finite inverse semigroups

We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial rook monoid admits no finite identity basis, and so do almost all additively idempotent semirings whose multiplicative reducts are combinatorial inverse semigroups.

math.GR↗