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.

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 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.

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 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.

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 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 Generators and relations of Rees matrix semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 481-495 ◽  
Author(s):  
H. Ayik ◽  
N. Ruškuc
Keyword(s):  
Rees Matrix Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Generators And Relations ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroup ◽  
Finitely Presented ◽  
Matrix Semigroups ◽  
The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

scholarly journals Inverse semigroups whose full inverse subsemigroups form a chain

1981 ◽  
Vol 22 (2) ◽  
pp. 159-165 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Analogous Problem ◽  
A Chain ◽  
Inverse Subsemigroups

The structure of semigroups whose subsemigroups form a chain under inclusion was determined by Tamura [9]. If we consider the analogous problem for inverse semigroups it is immediate that (since idempotents are singleton inverse subsemigroups) any inverse semigroup whose inverse subsemigroups form a chain is a group. We will therefore, continuing the approach of [5, 6], consider inverse semigroups whose full inverse subsemigroups form a chain: we call these inverse ▽-semigroups.

scholarly journals Finite presentability of arithmetic groups over global function fields

1987 ◽  
Vol 30 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Helmut Behr
Keyword(s):  
Small Groups ◽  
Function Fields ◽  
Algebraic Groups ◽  
Number Fields ◽  
Arithmetic Groups ◽  
Finitely Generated ◽  
Global Function ◽  
Global Function Fields ◽  
Finite Presentability ◽  
Finitely Presented

Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but over global function fields this is not always true. All known exceptions are “small” groups, which means that either the rank of the algebraic group or the set S of the underlying S-arithmetic ring has to be small. There exists now a complete list of all such groups which are not finitely generated, whereas we onlyhave a conjecture which groups are finitely generated but not finitely presented.

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