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

Green's relations and stability for subsemigroups

We prove new results on inheritance of Green's relations by subsemigroups in the presence of stability of elements. We provide counterexamples in other cases to show in particular that not all right-stable semigroups are embeddable in left-stable semigroups. This is carried out in the context of a survey of the various closely related notions of stability and minimality of Green's classes that have appeared in the literature over the last sixty years, and which have sometimes been presented in different forms

Cross-connections of linear transformation semigroups

Cross-connection theory developed by Nambooripad is the construction of a regular semigroup from its principal left (right) ideals using categories. We use the cross-connection theory to study the structure of the semigroup Sing(V) of singular linear transformations on an arbitrary vector space V over a field K. There is an inbuilt notion of duality in the cross-connection theory, and we observe that it coincides with the conventional algebraic duality of vector spaces. We describe various cross-connections between these categories and show that although there are many cross-connections, upto isomorphism, we have only one semigroup arising from these categories. But if we restrict the categories suitably, we can construct some interesting subsemigroups of the variants of the linear transformation semigroup. © 2018, Springer Science+Business Media, LLC, part of Springer Nature

Every group is a maximal subgroup of the free idempotent generated semigroup over a band

Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area.PostprintPeer reviewe