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).

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.

Direct products of finite monogenic inverse semigroups

1982 ◽  
Vol 92 (3-4) ◽  
pp. 301-317 ◽  
Author(s):  
D. C. Trueman
Keyword(s):  
Inverse Semigroup ◽  
Direct Product ◽  
Finite Semigroup ◽  
Inverse Semigroups ◽  
Direct Products

SynopsisWe characterize direct products of finite monogenic inverse semigroups; we show that a finite monogenic inverse semigroup which is not a group is directly indecomposable and that a finite semigroup which is decomposable into a direct product of monogenic inverse semigroups which are not groups is uniquely so decomposable. We determine when a finite semigroup can be decomposed into a direct product of non-group monogenic inverse semigroups and show how the direct factors, if they exist, can be found.

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 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

scholarly journals ON THE PARTITION MONOID AND SOME RELATED SEMIGROUPS

2010 ◽  
Vol 83 (2) ◽  
pp. 273-288 ◽  
Author(s):  
D. G. FITZGERALD ◽  
KWOK WAI LAU
Keyword(s):  
Transformation Semigroup ◽  
Galois Connection ◽  
Inverse Semigroups ◽  
Natural Order ◽  
Binary Relations ◽  
Ideal Lattices ◽  
New Interpretation ◽  
Semigroup Of Transformations ◽  
The Ideal

AbstractThe partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

scholarly journals Embedding inverse semigroups of homeomorphisms on closed subsets

1977 ◽  
Vol 18 (2) ◽  
pp. 199-207 ◽  
Author(s):  
Bridget Bos Baird
Keyword(s):  
Topological Space ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Countable Space ◽  
The One ◽  
One Point Compactification ◽  
First Countable Space

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.

Inverse semigroups generated by a pair of subgroups

1977 ◽  
Vol 77 (1-2) ◽  
pp. 9-22 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Homomorphic Image ◽  
Main Part ◽  
Structure Theorem ◽  
Inverse Semigroups ◽  
Infinite Cyclic Group ◽  
The Third ◽  
Group A ◽  
Maximum Group

SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

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