Tight inverse semigroups that are not bisimple

The algebraic theory of inverse semigroups has received considerable attention in recent years with satisfactory structure theorems being obtained for extensive classes of these semigroups. As is the case with so much of semigroup theory, many of these structure theorems were obtained by, or had their origin in fundamental papers of, A. H. Clifford. For example, the structure of inverse semigroups with central idempotents (semilattices of groups) was obtained by Clifford in [1], that of Brandt semigroups was obtained in [2], while, in [3], Clifford described bisimple inverse semigroups in terms of their right unit subsemigroup. The third of these papers has been the basis for much of the research on structural aspects of inverse semigroups. On the one hand, its direct descendants include Reilly's papers on bi-simple inverse semigroups in terms of RP-systems [10], [11], and the author's description of O-bisimple inverse semigroups in terms of groups and semilattices [5]. On the other hand, it has led to Reilly's description [9] of w-bisimple inverse semigroups and Warne's characterisations of various classes of bisimple inverse semigroups with special semilattices of idempotents. These, in turn, have led to the determination of simple inverse semigroups with special semilattices of idempotents.

The structure of a bisimple inverse semigroup with an identity has been related by Clifford [2] to that of its right unit subsemigroup. In this paper we give an explicit structure theorem for bisimple inverse semigroups in which the idempotents form a simple descending chaine0 > e1 > e2.…We call such a semigroup a bisimple co-semigroup. The structure of a semigroup of this kind is shown to be determined entirely by its group of units and an endomorphism of its group of units.

A remark concerning congruences on [0-]bisimple inverse semigroups

In [1] Clifford showed that the structure of any bisimple inverse semigroup with identity is uniquely determined by that of its right unit subsemigroup. The object of this paper is to show that the structure of any bisimple inverse semigroup with or without identity is determined by that of any of its a-classes. Let us define a right partial semigroup S to be a set S together with a partial binary operation satisfying the following condition: (A) if, for elements a, b, c of S, a(bc) is defined then so also is (ab)c defined and then a(bc) = (ab)c. We say that a right partial semigroup S is isomorphic with a right partial semigroup T if there exists a bijection b of S onto T such that ab is defined if and only if aq bq is defined and such that if ab is defined then (ab)k = aq bb. We define an RP-system (R, P) to be a right partial semigroup R together with a subsemigroup P of R such that: P(1) ab is defined if and only if a E P, for all a, b in R; P(2) R has a left identity contained in P; P(3) ac = bc implies that a = b for all a, b E P, c E R; P(4) foralla,be R,PanPb=Pcforsomece R. It then follows from P(1) and P(3) that any left identity of R contained in P is, in fact, a two-sided identity for P and so is unique. Now consider any a-class R of a bisimple inverse semigroup S. If we define the partial binary operation o in R by the rule that a o b = ab if and only if ab e R, then with respect to this operation R is a right partial semigroup with a subsemigroup P such that (R, P) is an RP-system (Theorem 1.4). Note 1. RP-systems can be obtained from systems having products more generally defined than stipulated in P(1). We then just ignore products which do not satisfy the condition P(1) (cf. Examples 1, 2 in ?6). Note 2. In particular, R could be a lattice ordered group and P the positive cone of R. We show that for any RP-system (R, P) there exists a bisimple inverse semigroup, which we denote by R-1 o R, some a-class of which is isomorphic with R. Conversely, for any a-class R of a bisimple inverse semigroup S there is a subsemigroup P of R such that (R, P) is an RP-system and R1 o R is isomorphic with S.

On a Class of Lattice Ordered Inverse Semigroups

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s $${\mathcal {J}}$$ –relation. The corresponding equivalence relation $$\simeq _N$$ is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.

A partial automorphism of a semigroup $S$ is any isomorphism between its subsemigroups, and the set all partial automorphisms of $S$ with respect to composition is the inverse monoid called the partial automorphism monoid of $S$. Two semigroups are said to be $\cPA$-isomorphic if their partial automorphism monoids are isomorphic. A class $\K$ of semigroups is called $\cPA$-closed if it contains every semigroup $\cPA$-isomorphic to some semigroup from $\K$. Although the class of all inverse semigroups is not $\cPA$-closed, we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is $\cPA$-closed. It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is $\cPA$-closed. A semigroup is called $\cPA$-determined if it is isomorphic or anti-isomorphic to any semigroup that is $\cPA$-isomorphic to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are $\cPA$-determined.

We study some aspects of Schein’s theory of cosets for closed inverse subsemigroups of inverse semigroups. We establish an index formula for chains of subsemigroups, and an analogue of M. Hall’s Theorem on the number of cosets of a fixed finite index. We then investigate the relationships between the following properties of a closed inverse submonoid of an inverse monoid: having finite index; being a recognizable subset; being a rational subset; being finitely generated (as a closed inverse submonoid). A remarkable result of Margolis and Meakin shows that these properties are equivalent for a closed inverse submonoid of a free inverse monoid.

E-unitary inverse monoids and the Cayley graph of a group presentation

The purpose of this paper is to give a structure for a semigroup which is a semilattice of bisimple inverse semigroups and satisfies certain conditions. For such a semigroup, we characterize the idempotent separating congruences.

On ∼ bisimple right type B [formula omitted] semigroups as model of some fractal structure

On F-inverse covers of inverse monoids

The word problem for inverse monoids presented by one idempotent relator

Formations and i-Fitting classes of inverse semigroups, congruences and languages

This paper provides an abstract characterization of the inverse monoids that appear as monoids of bi-congruences of finite minimal algebras generating arithmetical varieties. As a tool, a matrix construction is introduced which might be of independent interest in inverse semigroup theory. Using this construction as well as Ramsey's theorem, we embed a certain kind of inverse monoid into a factorizable monoid of the same kind. As noticed by M. Lawson, this embedding entails that the embedded finite monoids have finite F-unitary cover.