scholarly journals Ideals of free inverse semigroups

1980 ◽  
Vol 30 (2) ◽  
pp. 157-167
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Proper Ideal ◽  
Principal Ideals ◽  
Free Inverse Semigroup ◽  
Necessary And Sufficient

AbstractIt is shown that no proper ideal of a free inverse semigroup is free and that every isomorphism between ideals is induced by a unique automorphism of the whole semigroup. In addition, necessary and sufficient conditions are given for two principal ideals to be isomorhic.

scholarly journals A simplification of D'Alarcao's idempotent separating extensions of inverse semigroups

1978 ◽  
Vol 1 (3) ◽  
pp. 393-396
Author(s):  
Constance C. Edwards
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Necessary And Sufficient

In [2] D'Alarcao states necessary and sufficient conditions for the attainment of an idempotent-separating extension of an inverse semigroup. To do this D'Alarcao needed essentially three mappings satisfying thirteen conditions. In this paper we show that one can achieve the same results with two mappings satisfying eight conditions.

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.

On the Structure of Inverse Semigroup Amalgams

1997 ◽  
Vol 07 (05) ◽  
pp. 577-604 ◽  
Author(s):  
Paul Bennett
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Maximal Subgroups ◽  
Regular Semigroups ◽  
Endomorphism Monoids ◽  
Amalgamated Free Product ◽  
Necessary And Sufficient

This paper is the second of two papers devoted to the study of amalgamated free products of inverse semigroups. We use the characterization of the Schützenberger automata given previously by the author to obtain structural results and preservational properties of lower bounded amalgams. Haataja, Margolis and Meakin have shown that if [S1,S2;U is an amalgam of regular semigroups in which S1∩ S2=U is a full regular subsemigroup of S1 and S2, then the maximal subgroups of the amalgamated free product S1*U S2 may be described by the fundamental groups of certain bipartite graphs of groups. In this paper we show that the maximal subgroups of a lower bounded amalgam [S1,S2;U] are either isomorphic copies of subgroups of S1 and S2 or can be described by the same Bass-Serre theory characterization. It follows, as for the regular case, that if S1 and S2 are combinatorial, then the maximal subgroups of S1*U S2 are free. By studying the endomorphism monoids of the Schützenberger graphs we obtain a number of results concerning when inverse semigroup properties are preserved under the amalgamated free product construction. For example, necessary and sufficient conditions are given for S1*U S2 to be completely semisimple. Under a mild assumption we establish necessary and sufficient conditions for S1*U S2 to have finite ℛ-classes. This enables us to reprove a result of Cherubini, Meakin and Piochi on amalgams of free inverse semigroups. Finally we give sufficient conditions for S1*U S2 to be E-unitary.

scholarly journals The Structure of 0-E-unitary inverse semigroups I: the monoid case

1999 ◽  
Vol 42 (3) ◽  
pp. 497-520 ◽  
Author(s):  
Mark V. Lawson
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Structure Theory ◽  
Inverse Monoids ◽  
Necessary And Sufficient

This is the first of three papers in which we generalise the classical McAlister structure theory for E-unitary inverse semigroups to those 0-E-unitary inverse semigroups which admit a 0-restricted, idempotent pure prehomomorphism to a primitive inverse semigroup. In this paper, we concentrate on finding necessary and sufficient conditions for the existence of such prehomomorphisms in the case of 0-E-unitary inverse monoids. A class of inverse monoids which satisfy our conditions automatically are those which are unambiguous except at zero, such as the polycyclic monoids.

A visit to maximal non-ACCP subrings

2015 ◽  
Vol 15 (01) ◽  
pp. 1650016 ◽  
Author(s):  
Noômen Jarboui ◽  
Manar El Islam Toumi
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Proper Ideal ◽  
Normal Pair ◽  
Necessary And Sufficient

In this paper, we characterize maximal non-ACCP subrings R of a domain S in case (R, S) is a residually algebraic pair and R is semilocal. In this paper, we also consider a K-algebra S, a nonzero proper ideal I of S and a subring D of the field K and we determine necessary and sufficient conditions in order that D + I is a maximal non-ACCP subring of S. This gives an example of a maximal non-ACCP subring R of a domain S such that (R, S) is a normal pair and R is not semilocal.

scholarly journals Free inverse semigroups

1973 ◽  
Vol 16 (4) ◽  
pp. 443-453 ◽  
Author(s):  
G. B. Preston
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Canonical Forms ◽  
Principal Ideals ◽  
Free Inverse Semigroup ◽  
Alternative Description ◽  
Alternative Procedures ◽  
The Way

In an important recent paper H. E. Scheiblich gave a construction of free inverse semigroups that throws considerable light on their structure [1]. In this note we give an alternative description of free inverse semigroups. What Scheiblich did was to construct a free inverse semigroup as a semigroup of isomorphisms between principal ideals of a semilattice E, say, thus realising free inverse semigroups as inversee subsemigroupss of the semigroup TE, a kind of inverse semigroup introduced and exploited by W. D. Munn [2]. We go instead directly to canonical forms for the elements of a free inverse semigroup. The connexion between our construction and that of Scheiblich's will be clear. There are several alternative procedures possible to reach our construction on which we comment on the way.

scholarly journals Free generators in free inverse semigroups

1972 ◽  
Vol 7 (3) ◽  
pp. 407-424 ◽  
Author(s):  
N.R. Reilly
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Idempotent Element ◽  
Necessary And Sufficient Condition ◽  
Countable Number ◽  
Number Of Generators ◽  
Free Inverse Semigroup ◽  
Free Generators ◽  
Necessary And Sufficient

Using the characterization of the free inverse semigroup F on a set X, given by Scheiblich, a necessary and sufficient condition is found for a subset K of an inverse semigroup S to be a set of free generators for the inverse sub semigroup of S generated by K. It is then shown that any non-idempotent element of F generates the free inverse semigroup on one generator and that if |X| > 2 then F contains the free inverse semigroup on a countable number of generators. In addition, it is shown that if |X| = 1 then F does not contain the free inverse semigroup on two generators.

BICYCLIC SUBSEMIGROUPS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

2010 ◽  
Vol 20 (01) ◽  
pp. 89-113 ◽  
Author(s):  
EMANUELE RODARO
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Inverse Semigroups ◽  
Amalgamated Free Product ◽  
Free Product With Amalgamation ◽  
Necessary And Sufficient ◽  
Completely Semisimple

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 characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max {|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the [Formula: see text]-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the [Formula: see text]-classes to be finite.

scholarly journals Boolean Zero Square (BZS) Semigroups

2021 ◽  
Vol 26 (1) ◽  
pp. 31-39
Author(s):  
Pinto G.A.
Keyword(s):  
Inverse Semigroup ◽  
Simple Semigroup ◽  
Sufficient Conditions ◽  
Zero Element ◽  
Necessary And Sufficient Conditions ◽  
Equivalence Relations ◽  
New Class ◽  
Necessary And Sufficient

We introduce a new class of semigroups, that we call BZS - Boolean Zero Square-semigroups. A semigroup S with a zero element, 0, is said to be a BZS semigroup if, for every , we have  or . We obtain some properties that describe the behaviour of the Green’s equivalence relations , ,  and . Necessary and sufficient conditions for a BZS semigroup to be a band and an inverse semigroup are obtained. A characterisation of a special type of BZS completely 0-simple semigroup is presented.

scholarly journals Trace functions on inverse semigroup algebras

1995 ◽  
Vol 52 (3) ◽  
pp. 359-372 ◽  
Author(s):  
D. Easdown ◽  
W.D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Semigroup Algebra ◽  
Inverse Semigroups ◽  
Trace Function ◽  
Complex Field ◽  
Semigroup Algebras ◽  
Complex Conjugation ◽  
Necessary And Sufficient ◽  
Completely Semisimple

Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and S is completely semisimple of a type that includes countable free inverse semigroups.

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