scholarly journals Rank properties in finite inverse semigroups

2000 ◽  
Vol 43 (3) ◽  
pp. 559-568 ◽  
Author(s):  
Maria Isabel ◽  
Marques Ribeiro
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Theory ◽  
Inverse Semigroups ◽  
Brandt Semigroup ◽  
Large Rank ◽  
Finite Set

AbstractTwo possible concepts of rank in inverse semigroup theory, the intermediate I-rank and the upper I-rank, are investigated for the finite aperiodic Brandt semigroup. The so-called large I-rank is found for an arbitrary finite Brandt semigroup, and the result is used to obtain the large rank of the inverse semigroup of all proper subpermutations of a finite set.

scholarly journals EMBEDDINGS IN COSET MONOIDS

2008 ◽  
Vol 85 (1) ◽  
pp. 75-80
Author(s):  
JAMES EAST
Keyword(s):  
Semigroup Forum ◽  
Inverse Semigroup ◽  
Simple Proof ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Simple Description ◽  
Group Of Units ◽  
Finite Monoids ◽  
Finite Set ◽  
Symmetric Inverse Semigroup

AbstractA submonoid S of a monoid M is said to be cofull if it contains the group of units of M. We extract from the work of Easdown, East and FitzGerald (2002) a sufficient condition for a monoid to embed as a cofull submonoid of the coset monoid of its group of units, and show further that this condition is necessary. This yields a simple description of the class of finite monoids which embed in the coset monoids of their group of units. We apply our results to give a simple proof of the result of McAlister [D. B. McAlister, ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum20 (1980), 255–267] which states that the symmetric inverse semigroup on a finite set X does not embed in the coset monoid of the symmetric group on X. We also explore examples, which are necessarily infinite, of embeddings whose images are not cofull.

scholarly journals Homogeneity of inverse semigroups

2018 ◽  
Vol 28 (05) ◽  
pp. 837-875 ◽  
Author(s):  
Thomas Quinn-Gregson
Keyword(s):  
Inverse Semigroup ◽  
Finite Groups ◽  
Abelian Groups ◽  
Semigroup Theory ◽  
Inverse Semigroups ◽  
Finitely Generated ◽  
Homogeneous Groups ◽  
Inverse Subsemigroups

An inverse semigroup [Formula: see text] is a semigroup in which every element has a unique inverse in the sense of semigroup theory, that is, if [Formula: see text] then there exists a unique [Formula: see text] such that [Formula: see text] and [Formula: see text]. We say that a countable inverse semigroup [Formula: see text] is a homogeneous (inverse) semigroup if any isomorphism between finitely generated (inverse) subsemigroups of [Formula: see text] extends to an automorphism of [Formula: see text]. In this paper, we consider both these concepts of homogeneity for inverse semigroups, and show when they are equivalent. We also obtain certain classifications of homogeneous inverse semigroups, in particular periodic commutative inverse semigroups. Our results may be seen as extending both the classification of homogeneous semilattices and the classification of certain classes of homogeneous groups, such as homogeneous abelian groups and homogeneous finite groups.

Free Objects in Certain Varieties of Inverse Semigroups

1990 ◽  
Vol 42 (6) ◽  
pp. 1084-1097 ◽  
Author(s):  
S. W. Margolis ◽  
J. C. Meakin ◽  
J. B. Stephen
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Inverse Semigroups ◽  
Covering Space ◽  
Brandt Semigroup ◽  
Graphical Methods ◽  
Free Objects ◽  
Variety Of Groups

AbstractIn this paper it is shown how the graphical methods developed by Stephen for analyzing inverse semigroup presentations may be used to study varieties of inverse semigroups. In particular, these methods may be used to solve the word problem for the free objects in the variety of inverse semigroups generated by the five-element combinatorial Brandt semigroup and in the variety of inverse semigroups determined by laws of the form xn = xn + 1. Covering space methods are used to study the free objects in a variety of the form ∨ where is a variety of inverse semigroups and is the variety of groups.

scholarly journals A rank for right congruences on inverse semigroups

2007 ◽  
Vol 76 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Victoria Gould
Keyword(s):  
Inverse Semigroup ◽  
Maximal Subgroup ◽  
Inverse Semigroups ◽  
The Other ◽  
Brandt Semigroup ◽  
Group Congruence ◽  
Existentially Closed ◽  
Bicyclic Monoid ◽  
A Chain ◽  
Lattice Of Congruences

The S-rank (where ‘S’ abbreviates ‘sandwich’) of a right congruence ρ on a semigroup S is the Cantor-Bendixson rank of ρ in the lattice of right congruences ℛ of S with respect to a topology we call the finite type topology. If every ρ ϵ ℛ possesses S-rank, then S is ranked. It is known that every right Noetherian semigroup is ranked and every ranked inverse semigroup is weakly right Noetherian. Moreover, if S is ranked, then so is every maximal subgroup of S. We show that a Brandt semigroup 0(G, I) is ranked if and only if G is ranked and I is finite.We establish a correspondence between the lattice of congruences on a chain E, and the lattice of right congruences contained within the least group congruence on any inverse semigroup S with semilattice of idempotents E(S) ≅ E. Consequently we argue that the (inverse) bicyclic monoid B is not ranked; moreover, a ranked semigroup cannot contain a bicyclic -class. On the other hand, B is weakly right Noetherian, and possesses trivial (hence ranked) subgroups.Our notion of rank arose from considering stability properties of the theory Ts of existentially closed (right) S-sets over a right coherent monoid S. The property of right coherence guarantees that the existentially closed S-sets form an axiomatisable class. We argue that B is right coherent. As a consequence, it follows from known results that TB is a theory of B-sets that is superstable but not totally transcendental.

Extensions of One Primitive Inverse Semigroup by Another

1972 ◽  
Vol 24 (2) ◽  
pp. 270-278
Author(s):  
Janet E. Ault
Keyword(s):  
Inverse Semigroup ◽  
Finite Number ◽  
Primitive Idempotent ◽  
Inverse Semigroups ◽  
Ideal Extension ◽  
Brandt Semigroup ◽  
Abstract Characterization

Every inverse semigroup containing a primitive idempotent is an ideal extension of a primitive inverse semigroup by another inverse semigroup. Consequently, in developing the theory of inverse semigroups, it is natural to study ideal extensions of primitive inverse semigroups (cf. [3; 7]). Since the structure of any primitive inverse semigroup is known, an obvious type of ideal extension to consider is that of one primitive inverse semigroup by another. In this paper, we will construct all such extensions and give an abstract characterization of the resulting semigroup.The problem of extending one primitive inverse semigroup by another can be essentially reduced to that of extending one Brandt semigroup by another Brandt semigroup. The latter problem has been solved by Lallement and Petrich in [3] in case the first Brandt semigroup has only a finite number of idempotents.

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 Biorder-preserving coextensions of fundamental semigroups

1988 ◽  
Vol 31 (3) ◽  
pp. 463-467 ◽  
Author(s):  
David Easdown
Keyword(s):  
Group Theory ◽  
Semigroup Theory ◽  
Building Blocks ◽  
Inverse Semigroups ◽  
Extension Theory ◽  
Precise Definition ◽  
Simple Groups ◽  
Seminal Work ◽  
Fundamental Semigroups

In any extension theory for semigroups one must determine the basic building blocks and then discover how they fit together to create more complicated semigroups. For example, in group theory the basic building blocks are simple groups. In semigroup theory however there are several natural choices. One that has received considerable attention, particularly since the seminal work on inverse semigroups by Munn ([14, 15]), is the notion of a fundamental semigroup. A semigroup is called fundamental if it cannot be [shrunk] homomorphically without collapsing some of its idempotents (see below for a precise definition).

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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