scholarly journals Congruences on semigroups generated by injective nilpotent transformations

2006 ◽  
Vol 74 (3) ◽  
pp. 393-409 ◽  
Author(s):  
M. Paula O. Marques-Smith ◽  
R.P. Sullivan
Keyword(s):  
Inverse Semigroup ◽  
Symmetric Inverse Semigroup ◽  
Infinite Set

In 1987, Sullivan characterised the elements of the semigroup NI(X) generated by the nilpotents in I(X), the symmetric inverse semigroup on an infinite set X; and, in the same year, Gomes and Howie did the same for finite X. In 1999, Marques-Smith and Sullivan determined all the ideals of NI(X) for arbitrary X. In this paper, we use that work to describe all the congruences on NI(X).

scholarly journals Baer-Levi semigroups of partial transformations

2004 ◽  
Vol 69 (1) ◽  
pp. 87-106 ◽  
Author(s):  
Fernanda A. Pinto ◽  
R.P. Sullivan
Keyword(s):  
Inverse Semigroup ◽  
Cancellative Semigroup ◽  
Green’S Relations ◽  
Green's Relations ◽  
Algebraic Properties ◽  
Symmetric Inverse Semigroup ◽  
Infinite Set

Let X be an infinite set and suppose א0 ≤ q ≤ |X|. The Baer-Levi semigroup on X is the set of all injective ‘total’ transformations α: X → X such that |X\Xα| = q. It is known to be a right simple, right cancellative semigroup without idempotents, its automorphisms are “inner”, and some of its congruences are restrictions of Malcev congruences on I(X), the symmetric inverse semigroup on X. Here we consider algebraic properties of the semigroup consisting of all injective ‘partial’ transformations α of X such that |X\Xα| = q: in particular, we descried the ideals and Green's relations of it and some of its subsemigroups.

scholarly journals INJECTIVE TRANSFORMATIONS WITH EQUAL GAP AND DEFECT

2009 ◽  
Vol 79 (2) ◽  
pp. 327-336 ◽  
Author(s):  
JINTANA SANWONG ◽  
R. P. SULLIVAN
Keyword(s):  
Inverse Semigroup ◽  
Green’S Relations ◽  
Maximal Subsemigroups ◽  
Green's Relations ◽  
Algebraic Properties ◽  
Symmetric Inverse Semigroup ◽  
Infinite Set

AbstractSuppose that X is an infinite set and I(X) is the symmetric inverse semigroup defined on X. If α∈I(X), we let dom α and ran α denote the domain and range of α, respectively, and we say that g(α)=|X/dom α| and d(α)=|X/ran α| is the ‘gap’ and the ‘defect’ of α, respectively. In this paper, we study algebraic properties of the semigroup $A(X)=\{\alpha \in I(X)\mid g(\alpha )=d(\alpha )\}$. For example, we describe Green’s relations and ideals in A(X), and determine all maximal subsemigroups of A(X) when X is uncountable.

scholarly journals On Maximal Subsemigroups of Partial Baer-Levi Semigroups

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Boorapa Singha ◽  
Jintana Sanwong
Keyword(s):  
Inverse Semigroup ◽  
Maximal Subsemigroups ◽  
Symmetric Inverse Semigroup ◽  
Infinite Set

Suppose thatXis an infinite set with|X|≥q≥ℵ0andI(X)is the symmetric inverse semigroup defined onX. In 1984, Levi and Wood determined a class of maximal subsemigroupsMA(using certain subsetsAofX) of the Baer-Levi semigroupBL(q)={α∈I(X):domα=Xand|X∖Xα|=q}. Later, in 1995, Hotzel showed that there are many other classes of maximal subsemigroups ofBL(q), but these are far more complicated to describe. It is known thatBL(q)is a subsemigroup of the partial Baer-Levi semigroupPS(q)={α∈I(X):|X∖Xα|=q}. In this paper, we characterize all maximal subsemigroups ofPS(q)when|X|>q, and we extendMAto obtain maximal subsemigroups ofPS(q)when|X|=q.

scholarly journals Congruences induced by transitive representations of inverse semigroups

1987 ◽  
Vol 29 (1) ◽  
pp. 21-40 ◽  
Author(s):  
Mario Petrich ◽  
Stuart Rankin
Keyword(s):  
General Theory ◽  
Inverse Semigroup ◽  
Symmetric Group ◽  
Transitive Group ◽  
Inverse Semigroups ◽  
Group Representations ◽  
Symmetric Inverse Semigroup ◽  
The Right ◽  
Special Case

Transitive group representations have their analogue for inverse semigroups as discovered by Schein [7]. The right cosets in the group case find their counterpart in the right ω-cosets and the symmetric inverse semigroup plays the role of the symmetric group. The general theory developed by Schein admits a special case discovered independently by Ponizovskiǐ [4] and Reilly [5]. For a discussion of this topic, see [1, §7.3] and [2, Chapter IV].

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.

PARABOLIC SUBGROUPS AND CUSPIDAL REPRESENTATIONS OF FINITE MONOIDS

1991 ◽  
Vol 01 (01) ◽  
pp. 33-47 ◽  
Author(s):  
JAN OKNIŃSKI ◽  
MOHAN S. PUTCHA
Keyword(s):  
Inverse Semigroup ◽  
Finite Groups ◽  
Special Class ◽  
Semigroup Algebra ◽  
Parabolic Subgroups ◽  
Finite Monoids ◽  
Representations Of Finite Groups ◽  
Symmetric Inverse Semigroup ◽  
Complex Semigroup ◽  
Cuspidal Representations

This paper is mostly concerned with arbitrary finite monoids M with the complex semigroup algebra [Formula: see text] semisimple. Using the 1942 work of Clifford, we develop for these monoids a theory of cuspidal representations. Harish-Chandra's philosophy of cuspidal representations of finite groups can then be derived with an appropriate specialization. For [Formula: see text], we use Solomon's Hecke algebra to obtain a correspondence between the 'simple' representations of [Formula: see text] and the representations of the symmetric inverse semigroup. We also prove a semisimplicity theorem for a special class of finite monoids of the type which was earlier used by the authors to prove the semisimplicity of [Formula: see text].

scholarly journals On the semigroup of all partial fence-preserving injections on a finite set

2017 ◽  
Vol 16 (12) ◽  
pp. 1750223 ◽  
Author(s):  
Ilinka Dimitrova ◽  
Jörg Koppitz
Keyword(s):  
Inverse Semigroup ◽  
Transformation Formula ◽  
Green’S Relations ◽  
Generating Set ◽  
Regular Elements ◽  
Green's Relations ◽  
Finite Set ◽  
Symmetric Inverse Semigroup ◽  
Text Element ◽  
Element Set

For [Formula: see text], let [Formula: see text] be an [Formula: see text]-element set and let [Formula: see text] be a fence, also called a zigzag poset. As usual, we denote by [Formula: see text] the symmetric inverse semigroup on [Formula: see text]. We say that a transformation [Formula: see text] is fence-preserving if [Formula: see text] implies that [Formula: see text], for all [Formula: see text] in the domain of [Formula: see text]. In this paper, we study the semigroup [Formula: see text] of all partial fence-preserving injections of [Formula: see text] and its subsemigroup [Formula: see text]. Clearly, [Formula: see text] is an inverse semigroup and contains all regular elements of [Formula: see text] We characterize the Green’s relations for the semigroup [Formula: see text]. Further, we prove that the semigroup [Formula: see text] is generated by its elements with rank [Formula: see text]. Moreover, for [Formula: see text], we find the least generating set and calculate the rank of [Formula: see text].

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