Rings graded by bisimple inverse semigroups

2000 ◽  
Vol 130 (3) ◽  
pp. 603-609 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Maximal Subgroup ◽  
Sufficient Conditions ◽  
Inverse Semigroups ◽  
Semigroup Rings ◽  
Special Cases

Sufficient conditions are obtained for a ring R, faithfully graded by a bisimple inverse semigroup S, to be (a) prime and (b) right primitive, these conditions being on the subring RG consisting of all elements of R with support contained in G, a maximal subgroup of S. Earlier results on semigroup rings arise as special cases.

scholarly journals Chain conditions and semigroup graded rings

1988 ◽  
Vol 45 (3) ◽  
pp. 372-380 ◽  
Author(s):  
E. Jespers
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Cancellative Semigroup ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Graded Ring ◽  
Product Group ◽  
Chain Conditions ◽  
Unique Product

AbstractThe following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R[S] we also give a solution to the problem in case S is an inverse semigroup.

scholarly journals A simplification of D'Alarcao's idempotent separating extensions of inverse semigroups

1978 ◽  
Vol 1 (3) ◽  
pp. 393-396
Author(s):  
Constance C. Edwards
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Necessary And Sufficient

In [2] D'Alarcao states necessary and sufficient conditions for the attainment of an idempotent-separating extension of an inverse semigroup. To do this D'Alarcao needed essentially three mappings satisfying thirteen conditions. In this paper we show that one can achieve the same results with two mappings satisfying eight conditions.

Inverse semigroup homomorphisms via partial group actions

2001 ◽  
Vol 64 (1) ◽  
pp. 157-168 ◽  
Author(s):  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Group Actions ◽  
Inverse Semigroups ◽  
Inverse Monoid ◽  
Partial Group ◽  
Inverse Monoids ◽  
Maximal Group ◽  
Special Cases ◽  
Quick Proof ◽  
Group Component

This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.

scholarly journals Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings

2020 ◽  
pp. 1-35
Author(s):  
Daniel Gonçalves ◽  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Ring ◽  
Semigroup Rings ◽  
Convolution Algebra ◽  
Special Cases ◽  
Unital Rings ◽  
Étale Groupoid

Abstract Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

Some completely semisimple HNN-extensions of inverse semigroups

2019 ◽  
Vol 30 (02) ◽  
pp. 217-243
Author(s):  
Mohammed Abu Ayyash ◽  
Alessandra Cherubini
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Bicyclic Semigroup ◽  
Hnn Extensions ◽  
Necessary And Sufficient ◽  
Completely Semisimple

We give necessary and sufficient conditions in order that lower bounded HNN-extensions of inverse semigroups and HNN-extensions of finite inverse semigroups are completely semisimple semigroups. Since it is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup, we first characterize such HNN-extensions containing a bicyclic subsemigroup making use of the special feature of their Schützenberger automata.

BICYCLIC SUBSEMIGROUPS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

2010 ◽  
Vol 20 (01) ◽  
pp. 89-113 ◽  
Author(s):  
EMANUELE RODARO
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Inverse Semigroups ◽  
Amalgamated Free Product ◽  
Free Product With Amalgamation ◽  
Necessary And Sufficient ◽  
Completely Semisimple

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max {|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the [Formula: see text]-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the [Formula: see text]-classes to be finite.

scholarly journals TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

2021 ◽  
pp. 1-36
Author(s):  
BENJAMIN STEINBERG
Keyword(s):  
Inverse Semigroup ◽  
Cohomology Group ◽  
Group Structure ◽  
Inverse Semigroups ◽  
Crossed Product ◽  
Crossed Products ◽  
Semigroup Rings ◽  
Abelian Extensions ◽  
Algebraic Setting ◽  
Baer Sum

Abstract Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists. We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

On the Structure of Inverse Semigroup Amalgams

1997 ◽  
Vol 07 (05) ◽  
pp. 577-604 ◽  
Author(s):  
Paul Bennett
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Maximal Subgroups ◽  
Regular Semigroups ◽  
Endomorphism Monoids ◽  
Amalgamated Free Product ◽  
Necessary And Sufficient

This paper is the second of two papers devoted to the study of amalgamated free products of inverse semigroups. We use the characterization of the Schützenberger automata given previously by the author to obtain structural results and preservational properties of lower bounded amalgams. Haataja, Margolis and Meakin have shown that if [S1,S2;U is an amalgam of regular semigroups in which S1∩ S2=U is a full regular subsemigroup of S1 and S2, then the maximal subgroups of the amalgamated free product S1*U S2 may be described by the fundamental groups of certain bipartite graphs of groups. In this paper we show that the maximal subgroups of a lower bounded amalgam [S1,S2;U] are either isomorphic copies of subgroups of S1 and S2 or can be described by the same Bass-Serre theory characterization. It follows, as for the regular case, that if S1 and S2 are combinatorial, then the maximal subgroups of S1*U S2 are free. By studying the endomorphism monoids of the Schützenberger graphs we obtain a number of results concerning when inverse semigroup properties are preserved under the amalgamated free product construction. For example, necessary and sufficient conditions are given for S1*U S2 to be completely semisimple. Under a mild assumption we establish necessary and sufficient conditions for S1*U S2 to have finite ℛ-classes. This enables us to reprove a result of Cherubini, Meakin and Piochi on amalgams of free inverse semigroups. Finally we give sufficient conditions for S1*U S2 to be E-unitary.

A class of contracted inverse semigroup rings

1987 ◽  
Vol 107 (1-2) ◽  
pp. 175-196 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Finiteness Condition ◽  
Inverse Semigroups ◽  
Semigroup Rings ◽  
Chain Conditions ◽  
Direct Sum ◽  
The Given

SynopsisContracted inverse semigroup rings are studied subject to the restriction that the semilattices of the given inverse semigroups satisfy a certain finiteness condition, introduced in 1980 by Teply, Turman and Quesada. Results are obtained on semiprimitivity, primitivity, primeness, decomposition into a direct sum of ideals, and chain conditions on one-sided ideals.

scholarly journals Trace functions on inverse semigroup algebras

1995 ◽  
Vol 52 (3) ◽  
pp. 359-372 ◽  
Author(s):  
D. Easdown ◽  
W.D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Semigroup Algebra ◽  
Inverse Semigroups ◽  
Trace Function ◽  
Complex Field ◽  
Semigroup Algebras ◽  
Complex Conjugation ◽  
Necessary And Sufficient ◽  
Completely Semisimple

Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and S is completely semisimple of a type that includes countable free inverse semigroups.

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