Сложность проверки тождеств в полугруппах преобразований ранга 2

В статье исследуются полугруппы Т2(n), состоящие из всех преобразований n-элементного множества с не более чем 2-элементным образом

A minimal nonfinitely based semigroup whose variety is polynomially recognizable

We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.Comment: 16 pages, 3 figure

On the representation number of a crown graph

A graph G = (V,E) is word-representable if there exists a word ω over the alphabet V such that letters x and y alternate in ω if and only if xy is an edge in E . It is known (Kitaev and Pyatkin, 2008) that any word-representable graph G is k-word-representable for some k, that is, there exists a word ω representing G such that each letter occurs exactly k times in ω. The minimum such k is called G’s representation number. A crown graph (also known as a cocktail party graph) Hn,n is a graph obtained from the complete bipartite graph Kn,n by removing a perfect matching. In this paper, we show that for n≥ 5,Hn,n ’s representation number is [n / 2]. This result not only provides a complete solution to the open Problem 7.4.2 in Kitaev and Lozin (2015), but also gives a negative answer to the question raised in Problem 7.2.7 in Kitaev and Lozin (2015) on 3-word-representability of bipartite graphs. As a byproduct, we obtain a new example of a graph class with a high representation number

Сложность задачи проверки тождеств в одной конечной 0-простой полугруппе

Под задачей проверки тождеств ID-CHECK (А) в конечной алгебре А понимается комбинаторная задача распознавания, принимающая на вход пару терминов p=q и отвечающая на вопрос, выполнено ли тождество p=g в алгебре А. В работе исследуется вычислительная сложность задачи ID-CHECK в одной 19-элементной 0-простой полугруппе М. Доказана соNP-полнота задачи ID-CHECK (М), а также показано, что для всех меньших по числу элементов 0-простых полугрупп задача ID-CHECK полиномиальна. Кроме того, в работе исследуются две задачи о гомоморфизмах двудольных графов, для которых доказывается их соNP-полнота и проводится полиномиальная сводимость к ID-CHECK (М)

Hardness of Equations over Finite Solvable Groups Under the Exponential Time Hypothesis

Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in ? for nilpotent groups while it is ??-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of Fitting length two. In this work, we present the first lower bounds for the equation satisfiability problem in finite solvable groups: under the assumption of the exponential time hypothesis, we show that it cannot be in ? for any group of Fitting length at least four and for certain groups of Fitting length three. Moreover, the same hardness result applies to the equation identity problem

A Minimal Nonfinitely Based Semigroup Whose Variety is Polynomially Recognizable

We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm. © 2011 Springer Science+Business Media, Inc.Acknowledgement. The first and the second authors acknowledge support from the Federal Education Agency of Russia, project 2.1.1/3537, and from the Russian Foundation for Basic Research, grant 09-01-12142

Identities of the kauffman monoid K4 and of the Jones Monoid J4

Kauffman monoids Kn and Jones monoids Jn, n=2,3,…, are two families of monoids relevant in knot theory. We prove a somewhat counterintuitive result that the Kauffman monoids K3 and K4 satisfy exactly the same identities. This leads to a polynomial time algorithm to check whether a given identity holds in K4. As a byproduct, we also find a polynomial time algorithm for checking identities in the Jones monoid J4. © Springer Nature Switzerland AG 2020.M. V. Volkov—Supported by Ural Mathematical Center under agreement No. 075-02-2020-1537/1 with the Ministry of Science and Higher Education of the Russian Federation

From AA to BB to ZZ

The variety generated by the Brandt semigroup ${\bf B}_2$ can be defined within the variety generated by the semigroup ${\bf A}_2$ by the single identity $x^2y^2\approx y^2x^2$. Edmond Lee asked whether or not the same is true for the monoids ${\bf B}_2^1$ and ${\bf A}_2^1$. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of ${\bf A}_2^1$ that satisfy $x^2y^2\approx y^2x^2$ and contain ${\bf B}_2^1$. A further consequence is that the variety of ${\bf B}_2^1$ cannot be defined within the variety of ${\bf A}_2^1$ by any finite system of identities. Continuing downward, we then turn to subvarieties of ${\bf B}_2^1$. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity $x^2y\approx yx^2$ and containing the monoid $M({\bf z}_\infty)$, where ${\bf z}_\infty$ denotes the infinite limit of the Zimin words ${\bf z}_0=x_0$, ${\bf z}_{n+1}={\bf z}_n x_{n+1}{\bf z}_n$