scholarly journals Étale groupoids as germ groupoids and their base extensions

2010 ◽  
Vol 53 (3) ◽  
pp. 765-785 ◽  
Author(s):  
Dmitry Matsnev ◽  
Pedro Resende
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Inverse Monoids ◽  
Étale Groupoid

AbstractWe introduce the notion of wide representation of an inverse semigroup and prove that with a suitably defined topology there is a space of germs of such a representation that has the structure of an étale groupoid. This gives an elegant description of Paterson's universal groupoid and of the translation groupoid of Skandalis, Tu and Yu. In addition, we characterize the inverse semigroups that arise from groupoids, leading to a precise bijection between the class of étale groupoids and the class of complete and infinitely distributive inverse monoids equipped with suitable representations, and we explain the sense in which quantales and localic groupoids carry a generalization of this correspondence.

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 THE ÉTALE GROUPOID OF AN INVERSE SEMIGROUP AS A GROUPOID OF FILTERS

2013 ◽  
Vol 94 (2) ◽  
pp. 234-256 ◽  
Author(s):  
M. V. LAWSON ◽  
S. W. MARGOLIS ◽  
B. STEINBERG
Keyword(s):  
Functional Analysis ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Groupoid ◽  
Linear Representations ◽  
Inverse Subsemigroups ◽  
Étale Groupoid

AbstractPaterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz’s construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occurring in the associated topological groupoid.

scholarly journals Inverse semigroups determined by their partial automorphism monoids

2006 ◽  
Vol 81 (2) ◽  
pp. 185-198 ◽  
Author(s):  
Simon M. Goberstein
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Inverse Subsemigroups

AbstractThe partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup S with no isolated nontrivial subgroups is lattice determined ‘modulo semilattices’ and if T is an inverse semigroup whose partial automorphism monoid is isomorphic to that of S, then either S and T are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if T is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of S and T, respectively, are isomorphic. Moreover, for these results to hold, the conditions that S be tightly connected and have no isolated nontrivial subgroups are essential.

scholarly journals Expansions of inverse semigroups

2006 ◽  
Vol 80 (2) ◽  
pp. 205-228 ◽  
Author(s):  
Mark V. Lawson ◽  
Stuart W. Margolis ◽  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Inverse Monoids ◽  
New Class ◽  
Group Case

AbstractWe construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we term F-morphisms. These play the same role in the theory of idempotent-pure homomorphisms that F-inverse monoids play in the theory of E-unitary inverse semigroups.

scholarly journals The Structure of 0-E-unitary inverse semigroups I: the monoid case

1999 ◽  
Vol 42 (3) ◽  
pp. 497-520 ◽  
Author(s):  
Mark V. Lawson
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Structure Theory ◽  
Inverse Monoids ◽  
Necessary And Sufficient

This is the first of three papers in which we generalise the classical McAlister structure theory for E-unitary inverse semigroups to those 0-E-unitary inverse semigroups which admit a 0-restricted, idempotent pure prehomomorphism to a primitive inverse semigroup. In this paper, we concentrate on finding necessary and sufficient conditions for the existence of such prehomomorphisms in the case of 0-E-unitary inverse monoids. A class of inverse monoids which satisfy our conditions automatically are those which are unambiguous except at zero, such as the polycyclic monoids.

scholarly journals INVARIANT SETS AND NORMAL SUBGROUPOIDS OF UNIVERSAL ÉTALE GROUPOIDS INDUCED BY CONGRUENCES OF INVERSE SEMIGROUPS

2021 ◽  
pp. 1-20
Author(s):  
FUYUTA KOMURA
Keyword(s):  
Inverse Semigroup ◽  
Invariant Set ◽  
Fundamental Concept ◽  
Inverse Semigroups ◽  
Invariant Sets ◽  
Étale Groupoid

Abstract For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotients of inverse semigroups. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.

scholarly journals Simplicity of inverse semigroup and étale groupoid algebras

2021 ◽  
Vol 380 ◽  
pp. 107611
Author(s):  
Benjamin Steinberg ◽  
Nóra Szakács
Keyword(s):  
Inverse Semigroup ◽  
Étale Groupoid

scholarly journals A NOTE ON THE HOWSON PROPERTY IN INVERSE SEMIGROUPS

2016 ◽  
Vol 94 (3) ◽  
pp. 457-463 ◽  
Author(s):  
PETER R. JONES
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Necessary And Sufficient

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is again finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.

scholarly journals Inverse semigroups and their natural order

1978 ◽  
Vol 19 (1) ◽  
pp. 59-65 ◽  
Author(s):  
H. Mitsch
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Homomorphic Image ◽  
Point Of View ◽  
Inverse Semigroups ◽  
Natural Order ◽  
Theoretical Point ◽  
Trivial Group

The natural order of an inverse semigroup defined by a ≤ b ⇔ a′b = a′a has turned out to be of great importance in describing the structure of it. In this paper an order-theoretical point of view is adopted to characterise inverse semigroups. A complete description is given according to the type of partial order an arbitrary inverse semigroup S can possibly admit: a least element of (S, ≤) is shown to be the zero of (S, ·); the existence of a greatest element is equivalent to the fact, that (S, ·) is a semilattice; (S, ≤) is directed downwards, if and only if S admits only the trivial group-homomorphic image; (S, ≤) is totally ordered, if and only if for all a, b ∈ S, either ab = ba = a or ab = ba = b; a finite inverse semigroup is a lattice, if and only if it admits a greatest element. Finally formulas concerning the inverse of a supremum or an infimum, if it exists, are derived, and right-distributivity and left-distributivity of multiplication with respect to union and intersection are shown to be equivalent.

scholarly journals FACTORIZATION THEOREMS FOR MORPHISMS OF ORDERED GROUPOIDS AND INVERSE SEMIGROUPS

2001 ◽  
Vol 44 (3) ◽  
pp. 549-569 ◽  
Author(s):  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Classical Theory ◽  
Derived Category ◽  
Inverse Semigroups ◽  
Subject Classification

AbstractAdapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup homomorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson’s version of Ehresmann’s Maximum Enlargement Theorem, from which can be deduced the classical theory of idempotent-pure inverse semigroup homomorphisms and $E$-unitary inverse semigroups.AMS 2000 Mathematics subject classification: Primary 20M18; 20L05; 20M17

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