### $\mathcal{L}$-classes of inverse semigroups.

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In the paper we give a survey of rather new notions and results which generalize classical ones in the theory of braids. Among such notions are various inverse monoids of partial braids. We also observe presentations different from standard Artin presentation for generalizations of braids. Namely, we consider presentations with small number of generators, Sergiescu graph-presentations and Birman-Ko-Lee presentation. The work of V.~V.~Chaynikov on the word and conjugacy problems for the singular...

This paper continues the joint work with A. R. Medghalchi (2012) and the author’s recent work (2015). For an inverse semigroup S, it is shown that ${A}_{p}\left(S\right)$ has a bounded approximate identity if and only if l¹(S) is amenable (a generalization of Leptin’s theorem) and that A(S), the Fourier algebra of S, is operator amenable if and only if l¹(S) is amenable (a generalization of Ruan’s theorem).

The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.

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 $\mathscr{H}$ 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...