scholarly journals Rees matrix semigroups and the regular semidirect product

2005 ◽  
Vol 79 (1) ◽  
pp. 39-60
Author(s):  
K. Auinger ◽  
M. B. Szendrei
Keyword(s):  
Semidirect Product ◽  
Inverse Semigroups ◽  
Group Variety ◽  
Rees Matrix Semigroups ◽  
Completely Simple Semigroups ◽  
Matrix Semigroups ◽  
Completely Simple

AbstractA generalization of the Pastijn product is introduced so that, on the level of e-varieties and pseudoe-varieties, this product and the regular semidirect product by completely simple semigroups ‘almost always’ coincide. This is applied to give a model of the bifree objects in every e-variety formed as a regular semidirect product of a variety of inverse semigroups by a variety of completely simple semigroups that is not a group variety.

scholarly journals Flows on Classes of Regular Semigroups and Cauchy Categories

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Suha Ahmed Wazzan
Keyword(s):  
General Structure ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Regular Semigroups ◽  
Rees Matrix Semigroups ◽  
Completely Simple Semigroups ◽  
Special Case ◽  
Brandt Semigroups ◽  
Matrix Semigroups ◽  
Completely Simple

We consider the structure of the flow monoid for some classes of regular semigroups (which are special case of flows on categories) and for Cauchy categories. In detail, we characterize flows for Rees matrix semigroups, rectangular bands, and full transformation semigroups and also describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. In fact, we obtain a general structure for the flow monoids on Cauchy categories.

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.

scholarly journals On bisimple semigroups generated by a finite number of idempotents

1982 ◽  
Vol 33 (1) ◽  
pp. 92-101 ◽  
Author(s):  
Karl Byleen
Keyword(s):  
Finite Number ◽  
Partial Order ◽  
Natural Partial Order ◽  
Rees Matrix Semigroups ◽  
Matrix Semigroups ◽  
Completely Simple

AbstractNon-completely simple bisimple semigroups S which are generated by a finite number of idempotents are studied by means of Rees matrix semigroups over local submonoids eSe, e = e2 ∈ S. If under the natural partial order on the set Es of idempotents of such a semigroup S the sets ω(e) = {ƒ ∈ Es: ƒ ≤ e} for each e ∈ Es are well-ordered, then S is shown to contain a subsemigroup isomorphic to Sp4, the fundamental four-spiral semigroup. A non-completely simple hisimple semigroup is constructed which is generated by 5 idempotents but which does not contain a subsemigroup isomorphic to Sp4.

ALMOST FACTORIZABLE LOCALLY INVERSE SEMIGROUPS

2011 ◽  
Vol 21 (07) ◽  
pp. 1037-1052 ◽  
Author(s):  
MÁRIA B. SZENDREI
Keyword(s):  
Inverse Semigroups ◽  
Completely Simple Semigroups ◽  
Order Ideals ◽  
Locally Inverse Semigroups ◽  
Completely Simple

We introduce a notion of almost factorizability within the class of all locally inverse semigroups by requiring a property of order ideals, and we prove that the almost factorizable locally inverse semigroups are just the homomorphic images of Pastijn products of normal bands by completely simple semigroups.

scholarly journals Associativity of the regular semidirect product of existence varieties

2000 ◽  
Vol 69 (1) ◽  
pp. 85-115 ◽  
Author(s):  
Bernd Billhardt ◽  
Mária B. Szendrei
Keyword(s):  
Semidirect Product ◽  
Regular Semigroups ◽  
Completely Simple Semigroups ◽  
Completely Simple

AbstractThe associativity of the regular semidirect product of existence varieties introduced by Jones and Trotter was proved under certain condition by Reilly and Zhang. Here we estabilsh associativity in many new cases. Moreover, we prove that the regular semidirect product is right distributive with respect to the join operation. In particular, both associativity and right distriutivity yiel within the varieties of completely simple semigroups. Analogous results are obtainedj for e-pseudovarieties of finite regular semigroups.

Expansions of completely simple semigroups

2004 ◽  
Vol 41 (1) ◽  
pp. 39-58
Author(s):  
B. Billhardt
Keyword(s):  
Semidirect Product ◽  
Simple Semigroup ◽  
Completely Simple Semigroup ◽  
Completely Simple Semigroups ◽  
Group 6 ◽  
The Relationship ◽  
Completely Simple

For any completely simple semigroup C a regular expansion S(C) is constructed which is the Birget-Rhodes prefix expansion CPr if C is a group [6]. We show that our construction generalizes two important features of CPr. Moreover we embed S (C) into a restricted semidirect product of a semilattice by C and investigate the relationship to the expansion P(C), introduced by Meakin [14].

Bisimple Inverse Semigroups as Semigroups of Ordered Triples

1968 ◽  
Vol 20 ◽  
pp. 25-39 ◽  
Author(s):  
N. R. Reilly ◽  
A. H. Clifford
Keyword(s):  
Inverse Semigroups ◽  
Fixed Group ◽  
The Third ◽  
Completely Simple Semigroups ◽  
Completely Simple

In (8) and (13) it has been shown that certain bisimple inverse semigroups, called bisimple ω-semigroups and bisimple Z-semigroups, can be represented as semigroups of ordered triples. In these cases, two of the components of each triple are integers, and the third is drawn from a fixed group. This representation is analogous to that given by the theorem of Rees (1, p. 94) concerning completely simple semigroups, and shares the same advantages.

On existence varieties of locally inverse semigroups

1994 ◽  
Vol 115 (2) ◽  
pp. 197-217 ◽  
Author(s):  
K. Auinger ◽  
J. Doyle ◽  
P. R. Jones
Keyword(s):  
Deep Structure ◽  
Inverse Semigroups ◽  
Direct Products ◽  
Regular Semigroups ◽  
Completely Simple Semigroups ◽  
Locally Inverse Semigroup ◽  
Existence Variety ◽  
Locally Inverse Semigroups ◽  
Completely Simple

AbstractA locally inverse semigroup is a regular semigroup S with the property that eSe is inverse for each idempotent e of S. Motivated by natural examples such as inverse semigroups and completely simple semigroups, these semigroups have been the subject of deep structure-theoretic investigations. The class ℒ ℐ of locally inverse semigroups forms an existence variety (or e-variety): a class of regular semigroups closed under direct products, homomorphic images and regular subsemigroups. We consider the lattice ℒ(ℒℐ) of e-varieties of such semigroups. In particular we investigate the operations of taking meet and join with the e-variety CS of completely simple semigroups. An important consequence of our results is a determination of the join of CS with the e-variety of inverse semigroups – it comprises the E-solid locally inverse semigroups. It is shown, however, that not every e-variety of E-solid locally inverse semigroups is the join of completely simple and inverse e-varieties.

Completely simple semigroups: free products, free semigroups and varieties

1981 ◽  
Vol 88 (3-4) ◽  
pp. 293-313 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Matrix Representation ◽  
Maximal Subgroups ◽  
Lattice Of Varieties ◽  
Free Products ◽  
Completely Simple Semigroups ◽  
Countable Rank ◽  
Free Semigroups ◽  
Abelian Subgroups ◽  
And Inversion ◽  
Completely Simple

SynopsisThe class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.

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