Abstract Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors connection with an abstract theory of Cartan pairs rings. In this paper we show that extensions ample groupoids correspond a precise manner to Boolean inverse semigroups. particular, discrete twists over certain abelian semigroups, and they are classified Lausch’s second cohomology gr...

We present sufficient conditions for a unary semigroup variety to have no finite basis for its equational theory. In particular, we exhibit a 6-element involutory semigroup which is inherently non-finitely based as a unary semigroup. As applications we get several naturally arising unary semigroups without finite identity bases, for example: the semigroup of all complex 2 × 2-matrices endowed w...

We develop a general approach to the study of maximal nilpotent subsemigroups of finite semigroups. This approach can be used to recover many known classifications of maximal nilpotent subsemigroups, in particular, for the symmetric inverse semigroup, the symmetric semigroup, and the factor power of the symmetric group. We also apply this approach to obtain a classification of maximal nilpotent...

We investigate the computational complexity of various decision problems related to conjugacy in finite inverse semigroups. describe deterministic algorithms requiring logarithmic space for checking if two elements are conjugate full semigroup with respect notions conjugacy. prove following $$\textsf{PSPACE}$$ -complete: given generators an semigroup, (1) whether generated contains a idempotent...

In this paper we study the relation between module amenability of $theta$-Lau product $A×_theta B$ and that of Banach algebras $A, B$. We also discuss module biprojectivity of $A×theta B$. As a consequent we will see that for an inverse semigroup $S$, $l^1(S)×_theta l^1(S)$ is module amenable if and only if $S$ is amenable.