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 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 Inverse semigroups with certain types of partial automorphism monoids

1990 ◽  
Vol 32 (2) ◽  
pp. 189-195 ◽  
Author(s):  
Simon M. Goberstein
Keyword(s):  
Inverse Semigroup ◽  
General Expression ◽  
Inverse Semigroups ◽  
Inverse Monoid ◽  
Exceptional Groups ◽  
Completely Semisimple

AbstractFor an inverse semigroup S, the set of all isomorphisms betweeninverse subsemigroups of S is an inverse monoid under composition which is denoted by (S) and called the partial automorphism monoid of S. Kirkwood [7] and Libih [8] determined which groups have Clifford partial automorphism monoids. Here we investigate the structure of inverse semigroups whose partial automorphism monoids belong to certain other important classes of inverse semigroups. First of all, we describe (modulo so called “exceptional” groups) all inverse semigroups S such that (S) is completely semisimple. Secondly, for an inverse semigroup S, we find a convenient description of the greatest idempotent-separating congruence on (S), using a well-known general expression for this congruence due to Howie, and describe all those inverse semigroups whose partial automorphism monoids are fundamental.

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.

scholarly journals Unit regular inverse monoids and Cliford monoids

2018 ◽  
Vol 7 (2.13) ◽  
pp. 306
Author(s):  
Sreeja V K
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Group Element ◽  
Inverse Monoid ◽  
Completely Regular Semigroup ◽  
Clifford Semigroup ◽  
Inverse Monoids ◽  
Completely Regular ◽  
Group Of Units

Let S be a unit regular semigroup with group of units G = G(S) and semilattice of idempotents E = E(S). Then for every there is a such that Then both xu and ux are idempotents and we can write or .Thus every element of a unit regular inverse monoid is a product of a group element and an idempotent. It is evident that every L-class and every R-class contains exactly one idempotent where L and R are two of Greens relations. Since inverse monoids are R unipotent, every element of a unit regular inverse monoid can be written as s = eu where the idempotent part e is unique and u is a unit. A completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. A Clifford semigroup is a completely regular inverse semigroup. Characterization of unit regular inverse monoids in terms of the group of units and the semilattice of idempotents is a problem often attempted and in this direction we have studied the structure of unit regular inverse monoids and Clifford monoids. 

PARTIAL ACTIONS OF GROUPS

2004 ◽  
Vol 14 (01) ◽  
pp. 87-114 ◽  
Author(s):  
J. KELLENDONK ◽  
MARK V. LAWSON
Keyword(s):  
Group Theory ◽  
Inverse Semigroup ◽  
Group Action ◽  
Partial Function ◽  
Inverse Semigroups ◽  
Explicit Description ◽  
Partial Action ◽  
Partial Group

A partial action of a group G on a set X is a weakening of the usual notion of a group action: the function G×X→X that defines a group action is replaced by a partial function; in addition, the existence of g·(h·x) implies the existence of (gh)·x, but not necessarily conversely. Such partial actions are extremely widespread in mathematics, and the main aim of this paper is to prove two basic results concerning them. First, we obtain an explicit description of Exel's universal inverse semigroup [Formula: see text], which has the property that partial actions of the group G give rise to actions of the inverse semigroup [Formula: see text]. We apply this result to the theory of graph immersions. Second, we prove that each partial group action is the restriction of a universal global group action. We describe some applications of this result to group theory and the theory of E-unitary inverse semigroups.

scholarly journals GRAPH IMMERSIONS, INVERSE MONOIDS AND DECK TRANSFORMATIONS

2020 ◽  
pp. 1-19 ◽  
Author(s):  
CORBIN GROOTHUIS ◽  
JOHN MEAKIN
Keyword(s):  
Analogous Result ◽  
Group Actions ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Connected Graphs ◽  
Covering Map

If $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ is a covering map between connected graphs, and $H$ is the subgroup of $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is isomorphic to $N(H)/H$ , where $N(H)$ is the normalizer of $H$ in $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$ . We show that an entirely analogous result holds for immersions between connected graphs, where the subgroup $H$ is replaced by the closed inverse submonoid of the inverse monoid $L(\unicode[STIX]{x1D6E4},v)$ used to construct the immersion. We observe a relationship between group actions on graphs and deck transformations of graph immersions. We also show that a graph immersion $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ may be extended to a cover $g:\tilde{\unicode[STIX]{x1D6E5}}\rightarrow \unicode[STIX]{x1D6E4}$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$ .

A CHARACTERIZATION OF THE INVERSE MONOID OF BI-CONGRUENCES OF CERTAIN ALGEBRAS

2009 ◽  
Vol 19 (06) ◽  
pp. 791-808 ◽  
Author(s):  
KALLE KAARLI ◽  
LÁSZLÓ MÁRKI
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Theory ◽  
Independent Interest ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Abstract Characterization ◽  
Finite Monoids ◽  
Ramsey’S Theorem ◽  
Matrix Construction

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.

HNN EXTENSIONS OF SEMILATTICES

1999 ◽  
Vol 09 (05) ◽  
pp. 555-596 ◽  
Author(s):  
AKIHIRO YAMAMURA
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Inverse Semigroups ◽  
The Third ◽  
Maximal Group ◽  
Hnn Extensions ◽  
Solvable Word Problem ◽  
Hnn Extension ◽  
Subgroup Theorem ◽  
Solvable Word

The main purpose of this paper is to investigate properties of an HNN extension of a semilattice, to give its equivalent characterizations and to discuss similarities with free groups. An HNN extension of a semilattice is shown to be a universal object in a certain category and an F-inverse cover over a free group for every inverse semigroup in the category. We also show that a graph with respect to a certain subset of an HNN extension of a semilattice is a tree and that this property characterizes an HNN extension of a semilattice. Moreover, we look into three subclasses: the class of full HNN extensions of semilattices with an identity, the class of universally E-unitary inverse semigroups and the class of HNN extensions of finite semilattices. The first class consists of factorizable E-unitary inverse semigroups whose maximal group homomorphic images are free. We obtain a generalization of the Nielsen–Schreier subgroup theorem to this class. The second consists of inverse semigroups presented by relations on Dyck words. An inverse semigroup in the third class has a relatively easy finite presentation using a Dyck language and has solvable word problem.

scholarly journals Dual symmetric inverse monoids and representation theory

1998 ◽  
Vol 64 (3) ◽  
pp. 345-367 ◽  
Author(s):  
D. G. Fitzgerald ◽  
Jonathan Leech
Keyword(s):  
Representation Theory ◽  
Inverse Semigroup ◽  
The Other ◽  
Natural Order ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Dual Pairs ◽  
Representations Of Groups ◽  
Dual Symmetric Inverse Monoids ◽  
Symmetric Inverse Monoid

AbstractThere is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid Ix, that is, a monoid of partial one-to-one selfmaps of a set X. The present paper describes the structure of a categorical dual Ix* to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in Ix and Ix*, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.

A description of E-unitary inverse semigroups

1983 ◽  
Vol 95 (3-4) ◽  
pp. 239-242 ◽  
Author(s):  
Ross Wilkinson
Keyword(s):  
Inverse Semigroup ◽  
Direct Product ◽  
Inverse Semigroups ◽  
Maximal Group ◽  
Morphic Image ◽  
New Interpretation

SynopsisAn E-unitary inverse semigroup, S, has the property that, if x=S, and e2 = e=S, then (xe)2 = xe implies that x2 = x. As a consequence of this, we can see that S is an extension of its semilattice of idempotents, E, by its maximal group morphic image, G. Thus, following McAlister (1974), we attempt to describe S in terms of E and G. If we extend the semilattice E to a larger semilattice F, we are able to describe S in terms of a semi-direct product of F and G, giving a new interpretation to the approach of Schein (1975).

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