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 FREE ADEQUATE SEMIGROUPS

2011 ◽  
Vol 91 (3) ◽  
pp. 365-390 ◽  
Author(s):  
MARK KAMBITES
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Explicit Description ◽  
Finitely Generated ◽  
Free Objects ◽  
Ample Semigroup ◽  
Free Inverse Semigroup ◽  
Adequate Semigroup

AbstractWe give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial ‘folding’ operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are 𝒥-trivial and never finitely generated as semigroups, and that those which are finitely generated as (2,1,1)-algebras have decidable word problem.

On product varieties of inverse semigroups

1979 ◽  
Vol 28 (1) ◽  
pp. 107-119 ◽  
Author(s):  
J. L. Bales
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Variety ◽  
Inverse Semigroups ◽  
Variety Of Groups ◽  
Image Position ◽  
20 M 05

AbstractThis paper extends results on product varieties of groups to inverse semigroups. We show that if is a variety of groups and any inverse semigroup variety, then ∘ is a variety. We give a characterization of the identities of∘in terms of the identities of and ofWe show that if does not contain the variety of all groups then it has uncountably many supervarieties. Finally we show that ifis another variety of groups then Subject classifiaction (Amer. Math. Soc. (MOS) 1970): 20 M 05.

scholarly journals Context-freeness of the languages of Schützenberger automata of HNN-extensions of finite inverse semigroups

2016 ◽  
Vol 99 (113) ◽  
pp. 177-191
Author(s):  
Mohammed Ayyash ◽  
Emanuele Rodaro
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Polynomial Time ◽  
Inverse Semigroups ◽  
Free Graph ◽  
Hnn Extensions ◽  
Language L ◽  
Hnn Extension ◽  
Context Free ◽  
Standard Presentation

We prove that the Sch?tzenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we show that the word problem for an HNN-extension of a finite inverse semigroup S is decidable and lies in the complexity class of polynomial time problems.

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 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.

A finitely generated variety of combinatorial inverse semigroups having uncountably many subvarieties

2002 ◽  
Vol 132 (6) ◽  
pp. 1373-1394 ◽  
Author(s):  
Jiří Kad'ourek
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Finitely Generated ◽  
Moderate Size ◽  
Variety Of Groups ◽  
Finitely Generated Variety ◽  
Combinatorial Inverse Semigroup ◽  
The Continuum

A finite combinatorial inverse semigroup Θ of moderate size is presented such that the variety of combinatorial inverse semigroups generated by Θ possesses the following properties. The lattice of all subvarieties of this variety has the cardinality of the continuum. Moreover, this semigroup Θ, and hence also the variety it generates and its subvarieties, all have E-unitary covers over any non-trivial variety of groups. This indicates that the mentioned uncountable sublattice appears quite near the bottom of the lattice of all varieties of combinatorial inverse semigroups.

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.

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 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.

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