FAITHFUL FUNCTORS FROM CANCELLATIVE CATEGORIES TO CANCELLATIVE MONOIDS WITH AN APPLICATION TO ABUNDANT SEMIGROUPS

2005 ◽  
Vol 15 (04) ◽  
pp. 683-698 ◽  
Author(s):  
VICTORIA GOULD ◽  
MARK KAMBITES
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
The Other ◽  
Identity Matrix ◽  
Rees Matrix Semigroup ◽  
Cancellative Monoid ◽  
Faithful Functor ◽  
Ample Semigroup ◽  
Abundant Semigroups ◽  
Matrix Semigroup

We prove that any small cancellative category admits a faithful functor to a cancellative monoid. We use our result to show that any primitive ample semigroup is a full subsemigroup of a Rees matrix semigroup [Formula: see text] where M is a cancellative monoid and P is the identity matrix. On the other hand a consequence of a recent result of Steinberg is that it is undecidable whether a finite ample semigroup embeds as a full subsemigroup of an inverse semigroup.

scholarly journals REES MATRIX COVERS FOR A CLASS OF SEMIGROUPS WITH LOCALLY COMMUTING IDEMPOTENTS

2001 ◽  
Vol 44 (1) ◽  
pp. 173-186 ◽  
Author(s):  
Tanveer A. Khan ◽  
Mark V. Lawson
Keyword(s):  
Inverse Semigroup ◽  
Subject Classification ◽  
Rees Matrix Semigroup ◽  
Primary 20M10 ◽  
Locally Inverse Semigroup ◽  
Matrix Semigroup

AbstractMcAlister proved that every regular locally inverse semigroup can be covered by a regular Rees matrix semigroup over an inverse semigroup by means of a homomorphism which is locally an isomorphism. We generalize this result to the class of semigroups with local units whose local submonoids have commuting idempotents and possessing what we term a ‘McAlister sandwich function’.AMS 2000 Mathematics subject classification: Primary 20M10. Secondary 20M17

REES MATRIX COVERS FOR TIGHT ABUNDANT SEMIGROUPS

2010 ◽  
Vol 03 (03) ◽  
pp. 409-425
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum ◽  
Yongqian Zhu
Keyword(s):  
Rees Matrix Semigroup ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Regular Elements ◽  
Abundant Semigroups ◽  
Matrix Semigroup

Rees matrix covers for regular semigroups were first studied by McAlister in 1984. Lawson extended McAlister's results to abundant semigroups in 1987. We consider here a semigroup whose set of regular elements forms a subsemigroup, named tight semigroups. In this paper, it is proved that an abundant semigroup is tight and locally E-solid if and only if it is an F-local isomorphic image of an abundant Rees matrix semigroup [Formula: see text] over a tight E-solid abundant semigroup T, where the entries of the sandwich matrix P of [Formula: see text] are regular elements of T. Our results enrich the result of Lawson on Rees matrix covers for a class of abundant semigroups and extend the results of McAlister on Rees matrix covers for regular semigroups.

scholarly journals Certain congruences on a completely regular semigroup

1974 ◽  
Vol 15 (2) ◽  
pp. 109-120 ◽  
Author(s):  
Thomas L. Pirnot
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Group Congruence ◽  
Completely Regular Semigroup ◽  
Rees Matrix Semigroup ◽  
Semilattice Congruence ◽  
Completely Regular ◽  
Maximal Group ◽  
Minimum Group ◽  
Matrix Semigroup

Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences ρ on a completely regular semigroup S such that S/ρ is a semilattice of groups. We shall call such a congruence an SG-congruence.

REES MATRIX THEOREM FOR $\mathcal{D}^{(\ell)}$-SIMPLE STRONGLY RPP SEMIGROUPS

2008 ◽  
Vol 01 (02) ◽  
pp. 215-223 ◽  
Author(s):  
Xiaojiang Guo ◽  
Yuqi Guo ◽  
K. P. Shum
Keyword(s):  
Classical Theorem ◽  
Rees Matrix Semigroup ◽  
Cancellative Monoid ◽  
Principal Ideals ◽  
Strongly Rpp Semigroup ◽  
Matrix Semigroup ◽  
Left Cancellative Monoid

A semigroup S is called rpp if all right principal ideals of S, regarded as S1-systems, are projective. An rpp semigroup S is said to be strongly rpp if for any a ∈ S, there exists a unique idempotent e such that [Formula: see text] and a = ea. In this paper, we show that a [Formula: see text]-simple strongly rpp semigroup can be expressed by a Rees matrix semigroup over a left cancellative monoid and conversely. Our result generalizes the classical theorem of Rees in 1940 and also amplifies the Rees theorem in semigroup given by Lallement and Petrich in 1969.

Rees matrix semigroups over inverse semigroups

1986 ◽  
Vol 102 (1-2) ◽  
pp. 61-90 ◽  
Author(s):  
F. J. Pastijn ◽  
Mario Petrich
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Rees Matrix Semigroup ◽  
Regular Semigroups ◽  
Rees Matrix Semigroups ◽  
Special Cases ◽  
Matrix Semigroup ◽  
Matrix Semigroups ◽  
Completely Simple

SynopsisA Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

ON LOCALLY EHRESMANN SEMIGROUPS

2011 ◽  
Vol 10 (06) ◽  
pp. 1165-1186 ◽  
Author(s):  
XUEMING REN ◽  
DANDAN YANG ◽  
K. P. SHUM
Keyword(s):  
Inverse Semigroup ◽  
Rees Matrix Semigroup ◽  
Locally Inverse Semigroup ◽  
Matrix Semigroup ◽  
Ehresmann Semigroups

It was first proved by McAlister in 1983 that every locally inverse semigroup is a locally isomorphic image of a regular Rees matrix semigroup over an inverse semigroup and Lawson in 2000 further generalized this result to some special locally adequate semigroups. In this paper, we show how these results can be extended to a class of locally Ehresmann semigroups.

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.

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.

KNITTED SEMILATTIC STRUCTURE OF REGULAR ${\cal H}^\dag$-CYBER GROUPS

2011 ◽  
Vol 04 (03) ◽  
pp. 545-557
Author(s):  
Yu Su ◽  
Xiangzhi Kong
Keyword(s):  
Recent Result ◽  
Classical Theorem ◽  
Regular Semigroups ◽  
Abundant Semigroups ◽  
Structure Theorems ◽  
Semilattice Of Semigroups ◽  
Normal Formula

We consider regular [Formula: see text]-cyber groups in the class of [Formula: see text]-abundant semigroups. By using knitted semilattice of semigroups, we give some structure theorems for regular [Formula: see text]-cyber groups, right quasi-normal [Formula: see text]-cyber groups and normal [Formula: see text]-cyber groups. Our main result generalizes a classical theorem of Petrich- Reilly on normal cryptic groups from the class of regular semigroups to the class of generalized abundant semigroups and also entriches a recent result of Guo-Shum on left cyber groups.

On Gaussian Correlation Inequalities for Rectangles and Symmetric Sets

2002 ◽  
Vol 53 (3-4) ◽  
pp. 245-248
Author(s):  
Subir K. Bhandari ◽  
Ayanendranath Basu
Keyword(s):  
Recent Result ◽  
Convex Sets ◽  
Convex Set ◽  
Correlation Inequalities ◽  
The Other ◽  
Symmetric Convex ◽  
Complete Proof ◽  
Other Hand

Pitt's conjecture (1977) that P( A ∩ B) ≥ P( A) P( B) under the Nn (0, In) distribution of X, where A, B are symmetric convex sets in IRn still lacks a complete proof. This note establishes that the above result is true when A is a symmetric rectangle while B is any symmetric convex set, where A, B ∈ IRn. We give two different proofs of the result, the key component in the first one being a recent result by Hargé (1999). The second proof, on the other hand, is based on a rather old result of Šidák (1968), dating back a period before Pitt's conjecture.

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