scholarly journals Varieties of completely regular semigroups

1985 ◽  
Vol 38 (3) ◽  
pp. 372-393 ◽  
Author(s):  
Norman R. Reilly
Keyword(s):  
Subdirect Product ◽  
Variety Of Algebras ◽  
Regular Semigroups ◽  
Basis Of Identities ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Lattice Of Subvarieties ◽  
Completely Regular Semigroups ◽  
And Inversion ◽  
Completely Simple

AbstractIf CS(respectively, O) denotes the class of all completely simple semigroups (respectively, semigroups that are orthodox unions of groups) then CS(respectively, O) is a variety of algebras with respect to the operations of multiplication and inversion. The main result shows that the lattice of subvarieties of is a precisely determined subdirect product of the lattice of subvarieties of CSand the lattice of subvarieties of O. A basis of identities is obtained for any variety in terms of bases of identities for . Several operators on the lattice of subvarieties of are also introduced and studied.

scholarly journals CHARACTERIZING SOME COMPLETELY REGULAR SEMIGROUPS BY THEIR SUBSEMIGROUPS

2013 ◽  
Vol 94 (3) ◽  
pp. 397-416 ◽  
Author(s):  
MARIO PETRICH
Keyword(s):  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Completely Regular Semigroups ◽  
Completely Simple

AbstractWe consider several familiar varieties of completely regular semigroups such as groups and completely simple semigroups. For each of them, we characterize their members in terms of absence of certain kinds of subsemigroups, as well as absence of certain divisors, and in terms of a homomorphism of a concrete semigroup into the semigroup itself. For each of these varieties $ \mathcal{V} $ we determine minimal non-$ \mathcal{V} $ varieties, provide a basis for their identities, determine their join and give a basis for its identities. Most of this is complete; one of the items missing is a basis for identities for minimal nonlocal orthogroups. Three tables and a figure illustrate the results obtained.

scholarly journals A special sublattice of the congruence lattice of a regular semigroup

1997 ◽  
Vol 40 (3) ◽  
pp. 457-472 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Distributive Lattice ◽  
Regular Semigroup ◽  
Congruence Lattice ◽  
Regular Semigroups ◽  
Generators And Relations ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Special Cases ◽  
Completely Regular Semigroups ◽  
Completely Simple

Let S be a regular semigroup and be its congruence lattice. For ρ ∈ , we consider the sublattice Lρ of generated by the congruences pw where w ∈ {K, k, T, t}* and w has no subword of the form KT, TK, kt, tk. Here K, k, T, t are the operators on induced by the kernel and the trace relations on . We find explicitly the least lattice L whose homomorphic image is Lρ for all ρ ∈ and represent it as a distributive lattice in terms of generators and relations. We also consider special cases: bands of groups, E-unitary regular semigroups, completely simple semigroups, rectangular groups as well as varieties of completely regular semigroups.

Regular semigroups which are subdirect products of a band and a semilattice of groups

1973 ◽  
Vol 14 (1) ◽  
pp. 27-49 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Special Cases ◽  
Completely Regular Semigroups ◽  
Semilattice Of Semigroups ◽  
Subdirect Products ◽  
Completely Simple

In the study of the structure of regular semigroups, it is customary to impose several conditions restricting the behaviour of ideals, idempotents or elements. In a few instances, one may represent them as subdirect products of some much more restricted types of regular semigroups, e.g., completely (0-) simple semigroups, bands, semilattices, etc. In particular, studying the structure of completely regular semigroups, one quickly distinguishes certain special cases of interest when these semigroups are represented as semilattices of completely simple semigroups. In fact, this semilattice of semigroups may be built in a particular way, idempotents may form a subsemigroup, ℋ may be a congruence, and so on.

A conjecture for varieties of completely regular semigroups

2019 ◽  
Vol 69 (3) ◽  
pp. 541-556
Author(s):  
Mario Petrich
Keyword(s):  
Unary Operation ◽  
Maximal Subgroups ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Form Part ◽  
Completely Regular Semigroups ◽  
Completely Simple

Abstract The class 𝒞ℛ of completely regular semigroups considered with the unary operation of inversion within maximal subgroups forms a variety. The B-relation on the lattice ℒ(𝒞ℛ) of subvarieties of 𝒞ℛ identifies two varieties if they contain the same bands. Its classes are intervals with the set Δ of upper ends of these intervals. Canonical varieties form part of Δ. Previously we determined the sublattice Ψ of ℒ(𝒞ℛ) generated by the variety 𝒞𝒮 of completely simple semigroups and six canonical varieties. The conjecture is that the sublattice of ℒ(𝒞ℛ) generated by 𝒞𝒮 and canonical varieties follows the pattern of the structure of Ψ.

On singleton kernel classes in the lattice of varieties of completely regular semigroups

2019 ◽  
Vol 29 (08) ◽  
pp. 1383-1407 ◽  
Author(s):  
Jiří Kad’ourek
Keyword(s):  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Completely Regular Semigroups ◽  
Kernel Class ◽  
Completely Simple

In this paper, it is shown that, for every non-trivial variety [Formula: see text] of groups, the variety [Formula: see text] of all completely regular semigroups all of whose subgroups belong to [Formula: see text] is minimal in its kernel class in the lattice [Formula: see text] of all varieties of completely regular semigroups, and hence it constitutes, in fact, a singleton kernel class in the lattice [Formula: see text]. Even more generally, it is shown that, for every variety [Formula: see text] of completely simple semigroups which does not consist entirely of rectangular groups, the variety [Formula: see text] of all completely regular semigroups all of whose completely simple subsemigroups belong to [Formula: see text] is minimal in its kernel class in the lattice [Formula: see text], and hence it likewise constitutes a singleton kernel class in the mentioned lattice [Formula: see text].

scholarly journals Certain homomorphisms of the lattice of varieties of completely simple semigroups

1984 ◽  
Vol 37 (3) ◽  
pp. 287-306 ◽  
Author(s):  
Mario Petrich ◽  
Norman R. Reilly
Keyword(s):  
Subdirect Product ◽  
Direct Products ◽  
Lattice Of Varieties ◽  
Completely Simple Semigroups ◽  
Lattice Of Subvarieties ◽  
And Inversion ◽  
Completely Simple

AbstractCompletely simple semigroups form a variety, , of algebras with the operations of multiplication and inversion. It is known that the mapping , where is the variety of all groups, is an isomorphism of the lattice of all subvarieties of onto a subdirect product of the lattice of subvarieties of and the interval . We consider embeddings of into certain direct products on the above pattern with rectangular bands, rectangular groups and central completely simple semigroups in place of groups.

scholarly journals On free products of completely regular semigroups

1994 ◽  
Vol 56 (2) ◽  
pp. 212-231
Author(s):  
Peter R. Jones
Keyword(s):  
Free Product ◽  
General Theorem ◽  
Maximal Subgroups ◽  
Free Products ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Products Of Groups ◽  
Completely Regular Semigroups ◽  
Completely Simple

AbstractThe free product *CRSi of an arbitrary family of disjoint completely simple semigroups {Si}i∈i, within the variety CR of completely regular semigroups, is described by means of a theorem generalizing that of Kaďourek and Polák for free completely regular semigroups. A notable consequence of the description is that all maximal subgroups of *CRSi are free, except for those in the factors Si themselves. The general theorem simplifies in the case of free CR-products of groups and, in particular, free idempotent-generated completely regular semigroups.

Rees matrix seminearring

2022 ◽  
Author(s):  
Pavel Pal ◽  
Rajlaxmi Mukherjee ◽  
Manideepa Ghosh
Keyword(s):  
New York ◽  
Semigroup Theory ◽  
Coordinate Structure ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Additive Semigroup ◽  
Completely Regular Semigroups ◽  
Work Done ◽  
Similar Kind ◽  
Completely Simple

As a continuation of the work done in (R. Mukherjee (Pal), P. Pal and S. K. Sardar, On additively completely regular seminearrings, Commun. Algebra 45(12) (2017) 5111–5122), in this paper, our objective is to characterize left (right) completely simple seminearrings in terms of Rees Construction by generalizing the concept of Rees matrix semigroup (J. M. Howie, Fundamentals of Semigroup Theory (Clarendon Press, Oxford, 1995); M. Petrich and N. R. Reilly, Completely Regular Semigroups (Wiley, New York, 1999)) and that of Rees matrix semiring (M. K. Sen, S. K. Maity and H. J. Weinert, Completely simple semirings, Bull. Calcutta Math. Soc. 97 (2005) 163–172). In Rees theorem, a completely simple semigroup is coordinatized in such a way that each element can be seen to be a triplet which gives this abstract structure a much more simpler look. In this paper, we have been able to construct a similar kind of coordinate structure of a restricted class of left (right) completely simple seminearrings taking impetus from (M. P. Grillet, Semirings with a completely simple additive semigroup, J. Austral. Math. Soc. 20(Ser. A) (1975) 257–267, Theorem [Formula: see text] and (M. K. Sen, S. K. Maity and H. J. Weinert, Completely simple semirings, Bull. Calcutta Math. Soc. 97 (2005) 163–172, Theorem [Formula: see text]).

THE IDEMPOTENTS IN A PERIODIC SEMIGROUP

1996 ◽  
Vol 06 (05) ◽  
pp. 511-540 ◽  
Author(s):  
FRANCIS PASTIJN
Keyword(s):  
Congruence Lattice ◽  
Semigroup Variety ◽  
Equivalence Relations ◽  
Idempotent Algebra ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Periodic Semigroup ◽  
Lattice Of Subvarieties ◽  
Completely Regular Semigroups ◽  
The Continuum

Let [Formula: see text] be the semigroup variety determined by the identity xm=xm+k. For [Formula: see text] we define operations on the set E(S) of idempotents of S and thus obtain the idempotent algebra of S. For any subvariety [Formula: see text] of [Formula: see text] the idempotent algebras of the members of [Formula: see text] form a variety [Formula: see text] and [Formula: see text] yields a complete homomorphism of the lattice [Formula: see text] of subvarieties of [Formula: see text] onto the lattice [Formula: see text] of subvarieties of [Formula: see text]. The lattice [Formula: see text] contains a ∩-semilattice isomorphic to the ∩-semilattice [Formula: see text] of group varieties of exponent dividing k for every m≥1. In particular, for appropriate k, the lattice of subvarieties of the variety of all idempotent algebras of the completely regular semigroups over groups that belong to [Formula: see text] is of the power of the continuum. For any [Formula: see text], ρ→ρ|E(S) yields a complete homomorphism of the congruence lattice of S into the lattice of equivalence relations on E(S).

Embedding Regular Semigroups into Idempotent Generated Ones

2010 ◽  
Vol 17 (02) ◽  
pp. 229-240 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Close Relationship ◽  
Completely Regular Semigroups ◽  
Relationship Of ◽  
And Inversion ◽  
The Relationship

Any semigroup S can be embedded into a semigroup, denoted by ΨS, having some remarkable properties. For general semigroups there is a close relationship between local submonoids of S and of ΨS. For a number of usual semigroup properties [Formula: see text], we prove that S and ΨS simultaneously satisfy [Formula: see text] or not. For a regular semigroup S, the relationship of S and ΨS is even closer, especially regarding the natural partial order and Green's relations; in addition, every element of ΨS is a product of at most four idempotents. For completely regular semigroups S, the relationship of S and ΨS is still closer. On the lattice [Formula: see text] of varieties of completely regular semigroups [Formula: see text] regarded as algebras with multiplication and inversion, by means of ΨS, we define an operator, denoted by Ψ. We compare Ψ with some of the standard operators on [Formula: see text] and evaluate it on a small sublattice of [Formula: see text].

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