Let $A$ be a Banach algebra and $E$ be a Banach $A$-bimodule then $S = A oplus E$, the $l^1$-direct sum of $A$ and $E$ becomes a module extension Banach algebra when equipped with the algebras product $(a,x).(a^prime,x^prime)= (aa^prime, a.x^prime+ x.a^prime)$. In this paper, we investigate $triangle$-amenability for these Banach algebras and we show that for discrete inverse semigroup $S$ with...

A semigroup S is called left ample if it can be embedded in the symmetric inverse IX of partial bijections a non-empty set X such that image closed under unary operation α → αα⁻¹, where α⁻¹ IX. Right semigroups are defined dually. both and right ample. monoid (left, right) as semigroup. We observe dominion an subsemigroup coincides with generated by it. then determine dominions certain submonoi...

Given a commutative semigroup S with 0, where 0 is the unique singleton ideal, we associate a simple graph Γ(S), whose vertices are labeled with the nonzero elements in S. Two vertices in Γ(S) are adjacent if and only if the corresponding elements multiply to 0. The inverse problem, i.e., given an arbitrary simple graph, whether or not it can be associated to some commutative semigroup, has pro...

Recall that an open subset S of a (not necessarily Hausdorff) étale groupoid G is said to be a slice (sometimes also a G-set , or a bissection) if the domain and range maps are injective on S. The collection of all slices forms an inverse semigroup which has often been studied alongside the groupoid itself. Among other things the natural action of this semigroup on the unit space G highlights t...

The Brauer semigroup Bn of partitions of a 2n-element set into two-element subsets, can be generalized in various ways. For example we may consider arbitrary partitions to obtain the semigroup Cn. The visualization of elements of these semigroups as chips leads us to the Temperley-Lieb semigroup TLn and its analogue TLCn. We study combinatorial properties of these four semigroups. For example, ...

We investigate various groupoids associated to an arbitrary inverse semigroup with zero. show that the groupoid of filters respect natural partial order is isomorphic germs arising from standard action on space idempotent filters. also restriction this isomorphism tight and ultrafilters.

Suppose that X is an infinite set with |X| ≥ q ≥ א0 and I X is the symmetric inverse semigroup defined on X. In 1984, Levi and Wood determined a class of maximal subsemigroups MA using certain subsets A of X of the Baer-Levi semigroup BL q {α ∈ I X : dom α X and |X \ Xα| q}. Later, in 1995, Hotzel showed that there are many other classes of maximal subsemigroups of BL q , but these are far more...

For any amalgam (S, T; U) of inverse semigroups, it is shown that the natural partial order on S *u T, the (inverse semigroup) free product of S and T amalgamating U, has a simple form onSUT. In particular, it follows that the semilattice of 5 *u T is a bundled semilattice of the corresponding semilattice amalgam (E(S), E(T); E(U)); taken jointly with a result of Teruo Imaoka, this gives that t...

We examine, in a general setting, a notion of inverse semigroup of left quotients, which we call left I-quotients. This concept has appeared, and has been used, as far back as Clifford’s seminal work describing bisimple inverse monoids in terms of their right unit subsemigroups. As a consequence of our approach, we find a straightforward way of extending Clifford’s work to bisimple inverse semi...

Abstract We give algebraic characterisations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms subquotients its Boolean inverse semigroup open local bisections. It yields turn both semigroups with meets their Booleanisation.