Embedding inverse semigroups in coset semigroups

The setK(G) of all cosets X of a group G, modulo all subgroups of G, forms an inverse semigroup under the multiplication X*Y=smallest coset that constains XY. In this note we show that each inverse semigroup S can be embedded in some coset semigroupK(G). This follows from a result which shows that symmetric inverse semigroups can be embedded in the coset semigroups of suitable symmetric groups. We also give necessary and sufficient conditions on an inverse semigroup S in order that it should be isomorphic to someK(G). This research was supported by a grant from the National Science Foundation.     

Orthodox semigroups whose idempotents satisfy a certain identity

An orthodox semigroup S is called a left [right] inverse semigroup if the set of idempotents of S satisfies the identity xyx=xy [xyx=yx]. Bisimple left [right] inverse semigroups have been studied by Venkatesan [6]. In this paper, we clarify the structure of general left [right] inverse semigroups. Further, we also investigate the structure of orthodox semigroups whose idempotents satisfy the identity xyxzx=xyzx. In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup.     

Characters of finite inverse semigroups

For elements of a finite inverse semigroup, an equivalence relation called p-conugacy is introduced. It is proved that for any matrix representation of a finite inverse semigroup the values of the character of the representation are equal on p-conjugate elements. The number of inequivalent irreducible matrix representations of a finite inverse semigroup over the field of complex numbers is equal to the number of classes of p-conjugate elements.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 207–215, 1977.     

On Ehresmann semigroups

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.

     

Congruences on generalized inverse semigroups

A generalized inverse semigroup is a regular semigroup whose idempotents satisfy a permutation identity X₁ X₂...X_n=X_p1 X_p2...X_pn, where (P₁, P₂..., P_n) is a nontrivial permutation of (1, 2,..., n). Yamada [4] has given a complete classification of generalized inverse semigroups in terms of inverse semigroups, left normal bands, and right normal bands. In this paper we show that every congruence on a generalized inverse semigroup is uniquely determined by a congruence on its associated inverse semigroup, left normal band, and right normal band. A converse is also provided. This paper is extracted from the doctoral thesis of the author written at Monash University under the direction of Professor G. B. Preston. The research was carried out while the author held a Commonwealth Postgraduate Award.     

Left-invertible semigroups on Hilbert spaces

For strongly continuous semigroups on a Hilbert space, we present a short proof of the fact that the left inverse of a left-invertible semigroup can be chosen to be a semigroup as well. Furthermore, we show that this semigroup need not to be unique.     

Characterizations of Morita equivalent inverse semigroups

We prove that four different notions of Morita equivalence for inverse semigroups motivated by C^∗-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units.     

Morita equivalence of semigroups with local units

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.     

Free inverse semigroups

This note announces a characterization of the free inverse semigroup I on a non-empty set X.     

Semilattices of bisimple inverse semigroups

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.     

Toposes for semigroups: an invitation

This is an expository paper in which we explain the basic ideas of topos theory in connection with semigroup theory. We focus mainly on the classifying topos of an inverse semigroup or pseudogroup, and to some extent on creating a dictionary between the language of semigroups and topos theory. We begin with the algebraic theory having to do with inverse semigroups, and then turn to an analysis of pseudogroups using sheaf theory. Our work includes some new material on wide semigroup homomorphisms and their geometric morphisms.

     

On bases of identities of finite inverse semigroups with solvable subgroups

It is shown that every finite inverse semigroup having only solvable subgroups, viewed as a semigroup with the additional unary operation of inversion, has no finite basis of identities, unless it is a strict inverse semigroup.     

Inverse subsemigroups and classes of finite aperiodic semigroups

We prove a number of results related to finite semigroups and their inverse subsemigroups, including the following. (1) A finite semigroup is aperiodic if and only if it is a homomorphic image of a finite semigroup whose inverse subsemigroups are semilattices. (2) A finite inverse semigroup can be represented by order-preserving mappings on a chain if and only if it is a semilattice. Finally, we introduce the concept of pseudo-small quasivariety of finite semigroups, generalizing the concept of small variety.     

Note on a certain class of orthodox semigroups

This is a continuation and also a supplement of the previous papers [5], [6] and [8] concerning orthodox semigroups¹). In [8], it has been shown that a quasi-inverse semigroup is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup. In this paper, we present a structure theorem for quasi-inverse semigroups and some relevant matters.     

Frucht theorem for inverse semigroups

In the paper, the problem of representing a finite inverse semigroup by partial transformations of a graph is treated. The notions of weighted graph and its weighted partial isomorphisms are introduced. The main result is that any finite inverse semigroup is isomorphic to the semigroup of weighted partial isomorphisms of a weighted graph. This assertion is a natural generalization of the Frucht theorem for groups. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 246–251, February, 1997. This research was partially supported by the International Science Foundation under grant No. GSU 041049. Translated by A. I. Shtern     

Cayley color graphs of inverse semigroups and groupoids

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.     

On finite-dimensional representations of compact inverse semigroups

W.D. Munn proved that a finite-dimensional representation of an inverse semigroup is equivalent to a ?-representation (by partial isometries) if and only if it is bounded. This paper gives a new analytic proof that every finite-dimensional representation of a compact inverse semigroup is equivalent to a ?-representation. This will be the main result of this paper.     

Inverse semigroups on graphs

For each [directed] graph we construct an inverse semigroup. Our main application is a simple proof of the characterization of partially ordered sets ofJ-classes of finite semigroups, and some generalizations; our proof avoids using the inductive construction of the previous method by one of the authors [4]. For a connected graph in which each vertex has index at least two, our construction gives a congruence free inverse semigroup. In the final section we describe how a slight modification bf the construction yields the polycyclic monoids.     

Naturally ordered regular semigroups with an inverse monoid transversal

The notion of an inverse transversal of a regular semigroup is well-known. Here we investigate naturally ordered regular semigroups that have an inverse transversal. Such semigroups are necessarily locally inverse and the inverse transversal is a quasi-ideal. After considering various general properties that relate the imposed order to the natural order, we highlight the situation in which the inverse transversal is a monoid. The regularity of Green’s relations is also characterised. Finally, we determine the structure of a naturally ordered regular semigroup with an inverse monoid transversal.