Inverse semigroups form a variety of unary semigroups, that is, semigroups equipped with an additional unary operation, in this case a 7→ a−1. The theory of inverse semigroups is perhaps the best developed within semigroup theory, and relies on two factors: an inverse semigroup S is regular, and has semilattice of idempotents. Three major approaches to the structure of inverse semigroups have e...

in this paper we find sufficient conditions on primitive inverse topological semigroup s under which: the inversion inv : (h(s)) (h(s)) is continuous; we show that every topologically periodic countable compact primitive inverse topological semigroups with closed h-classes is topologically isomorphic to an orthogonal sum p i2= bi (gi) of topological brandt extensions bi (gi) of countably compac...

An inverse semigroup is a semigroup S such that for each x e S there exists a unique inverse x~ & S such that both xx~x=x and x~xx~=x~\ This condition is equivalent to S both being (von Neumann) regular and having all idempotents commute. The classic example of such a semigroup is the symmetric inverse semigroup Ix of all partial bijections on a set X under the standard composition of partial f...

Let R be a commutative ring and S a finite locally inverse semigroup. It is proved that the semigroup algebra R S is isomorphic to the direct product of Munn algebras M R GJ , mJ , nJ ;PJ with J ∈ S/J, where mJ is the number of R-classes in J , nJ the number of L-classes in J , and GJ a maximum subgroup of J . As applications, we obtain the sufficient and necessary conditions for the semigroup ...

It is known [l; 2] that every inverse semigroup 5 has a faithful representation as a semigroup of (1, l)-mappings of subsets of a set A into A. The set A may be taken as the set of elements of 5 and the (1, l)-mappings as mappings of principal left ideals of 5 onto principal left ideals of 5. If £ is the set of idempotents of 5 then there is also a representation of 5, not necessarily faithful,...

In [1] we introduced the concept of restricted representations for an inverse semigroup S and studied the restricted forms of some important Banach algebras on S. In this paper, we continue our study by considering the relation between the restricted positive definite functions and retricted representations. In particular, we prove restricted versions of the Godement’s characterization of the p...

We consider a version of the notion of F -inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid construction, very similar to the one of a transformation group. We discuss examples related to Toeplitz algebras on subsemigroups of discrete groups, to Cuntz-Kr...

We study inverse semigroup amalgams [S1, S2;U ], where S1 and S2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S1 and S2. We use a modified version of Bennett’s work on the structure of Schützenberger graphs of the R-classes of S1 ∗U S2 to state sufficient conditions for the amalgamated free products S1 ∗...

In this paper it is shown how the graphical methods developed by Stephen for analyzing inverse semigroup presentations may be used to study varieties of inverse semigroups. In particular, these methods may be used to solve the word problem for the free objects in the variety of inverse semigroups generated by the five-element combinatorial Brandt semigroup and in the variety of inverse semigrou...

We argue that weak containment is an appropriate notion of amenability for inverse semigroups. Given an inverse semigroup S and a homomorphism φ of S onto a group G, we show, under an assumption on ker(φ), that S has weak containment if and only if G is amenable and ker(φ) has weak containment. Using Fell bundle amenability, we find a related result for inverse semigroups with zero. We show tha...