HNN extensions of inverse semigroups with zero

2007 ◽  
Vol 142 (1) ◽  
pp. 25-39 ◽  
Author(s):  
E. R. DOMBI ◽  
N. D. GILBERT
Keyword(s):  
Normal Form ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Universal Group ◽  
Main Application ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Natural Way

AbstractWe study a construction of an HNN extension for inverse semigroups with zero. We prove a normal form for the elements of the universal group of an inverse semigroup that is categorical at zero, and use it to establish structural results for the universal group of an HNN extension. Our main application of the HNN construction is to show that graph inverse semigroups –including the polycyclic monoids –admit HNN decompositions in a natural way, and that this leads to concise presentations for them.

scholarly journals Locally full HNN extensions of inverse semigroups

2001 ◽  
Vol 70 (2) ◽  
pp. 235-272 ◽  
Author(s):  
Akihiro Yamamura
Keyword(s):  
Normal Form ◽  
Inverse Semigroup ◽  
Fundamental Group ◽  
Maximal Subgroup ◽  
Inverse Semigroups ◽  
Group Presentation ◽  
Graph Of Groups ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Normal Form Theorem

AbstractWe investigate a locally full HNN extension of an inverse semigroup. A normal form theorem is obtained and applied to the word problem. We construct a tree and show that a maximal subgroup of a locally full HNN extension acts on the tree without inversion. Bass-Serre theory is employed to obtain a group presentation of the maximal subgroup as a fundamental group of a certain graph of groups associated with the D-structure of the original semigroup.

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.

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.

FINITE PRESENTABILITY OF HNN EXTENSIONS OF INVERSE SEMIGROUPS

2005 ◽  
Vol 15 (03) ◽  
pp. 423-436 ◽  
Author(s):  
E. R. DOMBI ◽  
N. D. GILBERT ◽  
N. RUŠKUC
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Finitely Generated ◽  
Semigroup Presentation ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Finite Presentability ◽  
Order Ideals ◽  
Inverse Subsemigroups ◽  
Finitely Presented

HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely generated or finitely presented base is again finitely generated or finitely presented. Our main results depend upon properties of the [Formula: see text]-preorder in the associated subsemigroups. Let S be a finitely generated inverse semigroup and let U, V be inverse subsemigroups of S, isomorphic via φ: U → V, that are order ideals in S. We prove that the HNN extension S*U,φ is finitely generated if and only if U is finitely [Formula: see text]-dominated. If S is finitely presented, we give a necessary and suffcient condition for S*U,φ to be finitely presented. Here, in contrast to the theory of HNN extensions of groups, it is not necessary that U be finitely generated.

scholarly journals A correspondence between a class of monoids and self-similar group actions II

2015 ◽  
Vol 25 (04) ◽  
pp. 633-668
Author(s):  
Mark V. Lawson ◽  
Alistair R. Wallis
Keyword(s):  
Group Actions ◽  
Similar Group ◽  
Universal Group ◽  
Length Functions ◽  
Group Of Units ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Self Similar ◽  
Cancellative Monoids

The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa–Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych. In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be significant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.

Some completely semisimple HNN-extensions of inverse semigroups

2019 ◽  
Vol 30 (02) ◽  
pp. 217-243
Author(s):  
Mohammed Abu Ayyash ◽  
Alessandra Cherubini
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Bicyclic Semigroup ◽  
Hnn Extensions ◽  
Necessary And Sufficient ◽  
Completely Semisimple

We give necessary and sufficient conditions in order that lower bounded HNN-extensions of inverse semigroups and HNN-extensions of finite inverse semigroups are completely semisimple semigroups. Since it is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup, we first characterize such HNN-extensions containing a bicyclic subsemigroup making use of the special feature of their Schützenberger automata.

A new definition of conjugacy for semigroups

2018 ◽  
Vol 17 (02) ◽  
pp. 1850032 ◽  
Author(s):  
Janusz Konieczny
Keyword(s):  
Group Theory ◽  
Inverse Semigroup ◽  
Directed Graphs ◽  
Conjugacy Classes ◽  
Inverse Semigroups ◽  
Transformation Semigroups ◽  
Arbitrary Semigroup ◽  
Basic Transformation ◽  
Definition Of ◽  
Natural Way

The conjugacy relation plays an important role in group theory. If [Formula: see text] and [Formula: see text] are elements of a group [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] for some [Formula: see text]. The group conjugacy extends to inverse semigroups in a natural way: for [Formula: see text] and [Formula: see text] in an inverse semigroup [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] and [Formula: see text] for some [Formula: see text]. In this paper, we define a conjugacy for an arbitrary semigroup [Formula: see text] that reduces to the inverse semigroup conjugacy if [Formula: see text] is an inverse semigroup. (None of the existing notions of conjugacy for semigroups has this property.) We compare our new notion of conjugacy with existing definitions, characterize the conjugacy in basic transformation semigroups and their ideals using the representation of transformations as directed graphs, and determine the number of conjugacy classes in these semigroups.

Amalgams vs Yamamura's HNN-Extensions of Inverse Semigroups

2011 ◽  
Vol 18 (04) ◽  
pp. 647-657 ◽  
Author(s):  
Alessandra Cherubini ◽  
Emanuele Rodaro
Keyword(s):  
Free Product ◽  
Inverse Semigroups ◽  
Hnn Extensions ◽  
Amalgamated Free Product ◽  
Hnn Extension

We investigate the connections between amalgams and Yamamura's HNN-extensions of inverse semigroups. In particular, we prove that amalgams of inverse semigroups with an identity adjoint are quotient semigroups of some special Yamamura's HNN-extensions. As a consequence, we show how to obtain the Schützenberger graph of a word w with respect to the presentation of an amalgamated free product starting from the Schützenberger graph of a suitable word w′ with respect to the presentation of the associated HNN-extension by simply making V-quotients and deletions of edges.

HNN Extensions of Inverse Semigroups and Applications

1997 ◽  
Vol 07 (05) ◽  
pp. 605-624 ◽  
Author(s):  
Akihiro Yamamura
Keyword(s):  
Inverse Semigroups ◽  
General Definition ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Algorithmic Problems ◽  
New Construction ◽  
Definition Of ◽  
Inverse Subsemigroups ◽  
Finitely Presented ◽  
Markov Properties

Originally the concept of an HNN extension of a group was introduced by Higman, Neumann and Neumann in their study of embeddability of groups. Howie introduced the concept of HNN extensions of semigroups and showed embeddability in the case that the associated subsemigroups are unitary. On the other hand, T. E. Hall showed the embeddability of HNN extensions of inverse semigroups of a special type in his survey article on amalgamation of inverse semigroups. We introduce a more general definition of an HNN extension and show that free inverse semigroups and the bicyclic semigroup are HNN extensions of semilattices as examples of our new construction. We discuss weak HNN embeddability in several classes of semigroups and strong HNN embeddability in the class of inverse semigroups. One of our main purposes in the study of HNN extensions of inverse semigroups is to employ HNN extensions to examine some algorithmic problems. We prove the undecidability of Markov properties of finitely presented inverse semigroups using HNN extensions. This result was announced by Vazhenin in 1978, but no proof of it has been published to date. We also show undecidability of several non-Markov properties and discuss some undecidable problems on finitely generated inverse subsemigroups of finitely presented inverse semigroups.

scholarly journals GRÖBNER–SHIRSHOV BASES FOR FREE INVERSE SEMIGROUPS

2009 ◽  
Vol 19 (02) ◽  
pp. 129-143 ◽  
Author(s):  
L. A. BOKUT ◽  
YUQUN CHEN ◽  
XIANGUI ZHAO
Keyword(s):  
Normal Form ◽  
Inverse Semigroup ◽  
Normal Forms ◽  
Inverse Semigroups ◽  
Free Inverse Semigroup ◽  
New Construction

A new construction for free inverse semigroups was obtained by Poliakova and Schein in 2005. Based on their result, we find Gröbner–Shirshov bases for free inverse semigroups with respect to the deg-lex order of words. In particular, we give the (unique and shortest) normal forms in the classes of equivalent words of a free inverse semigroup together with the Gröbner–Shirshov algorithm to transform any word to its normal form.

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