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

Equations on the semídírect product of a finite semilattice by a -trivial monoid of height k,

Abstract: Let Com t , q denote the variety of finite monoids that satisfy the equations xy = yx and x t = x t+q . The variety Com 1,1 is the variety of finite semilattices also denoted by J 1 . In this paper, we consider the product variety J 1 *Com t,q generated by all semidirect products of the form M * N with M J 1 and N Com t,q . We give a complete sequence of equations for J 1 * Com t,q implying complete sequences of equations for J 1 * (Com A), J 1 * (Com G) and J 1 * Com, where Com denotes the variety of finite commutative monoids, A the variety of finite aperiodic monoids and G the variety of finite groups. Article: 1. Introduction Let Com t , q denote the variety of finite monoids that satisfy the equations xy = yx and x t = x t+q . The variety Com 1,1 is the variety of finite semilattices also denoted by J 1 . In this paper, we give an equational characterization of the product variety J 1 * Com t , q generated by all semidirect products of the form M * N with M J 1 and N Com t , q . Our results imply a complete sequence of equations for J 1 * (Com A), J 1 *(Com G) and J 1 *Com, where Com denotes the variety of finite commutative monoids, A the variety of finite aperiodic monoids and G the variety of finite groups. Pin Our results follow from versions of techniques used in particular by , Brzozowski and Simon Definitions and notations Let M and N be monoids. We say that M divides N and write M N if M is a morphic image of a submonoid of N . Note that the divisibility relation is transitive. An M-variety V is a family of finite monoids that satisfies the following two conditions

Characterizing pure, cryptic and Clifford inverse semigroups

summary:An inverse semigroup $S$ is pure if $e=e^2$, $a\in S$, $e<a$ implies $a^2=a$; it is cryptic if Green's relation $\mathcal {H}$ on $S$ is a congruence; it is a Clifford semigroup if it is a semillatice of groups. We characterize the pure ones by the absence of certain subsemigroups and a homomorphism from a concrete semigroup, and determine minimal nonpure varieties. Next we characterize the cryptic ones in terms of their group elements and also by a homomorphism of a semigroup constructed in the paper. We also characterize groups and Clifford semigroups in a similar way by means of divisors. The paper also contains characterizations of completely semisimple inverse and of combinatorial inverse semigroups in a similar manner. It ends with a description of minimal non-$\mathcal {V}$ varieties, for varieties $\mathcal {V}$ of inverse semigroups considered