Rees matrix seminearring

Work Done ◽

Similar Kind ◽

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]).

CHARACTERIZING SOME COMPLETELY REGULAR SEMIGROUPS BY THEIR SUBSEMIGROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788713000049 ◽

2013 ◽

Vol 94 (3) ◽

pp. 397-416 ◽

Cited By ~ 1

Author(s):

MARIO PETRICH

Keyword(s):

Regular Semigroups ◽

Completely Regular ◽

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.

A special sublattice of the congruence lattice of a regular semigroup

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500023944 ◽

1997 ◽

Vol 40 (3) ◽

pp. 457-472 ◽

Cited By ~ 2

Author(s):

Mario Petrich

Keyword(s):

Distributive Lattice ◽

Regular Semigroup ◽

Congruence Lattice ◽

Regular Semigroups ◽

Generators And Relations ◽

Completely Regular ◽

Special Cases ◽

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

Glasgow Mathematical Journal ◽

10.1017/s0017089500001701 ◽

1973 ◽

Vol 14 (1) ◽

pp. 27-49 ◽

Cited By ~ 12

Author(s):

Mario Petrich

Keyword(s):

Regular Semigroups ◽

Completely Regular ◽

Special Cases ◽

Semilattice Of Semigroups ◽

Subdirect Products ◽

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

Mathematica Slovaca ◽

10.1515/ms-2017-0246 ◽

2019 ◽

Vol 69 (3) ◽

pp. 541-556

Author(s):

Mario Petrich

Keyword(s):

Unary Operation ◽

Maximal Subgroups ◽

Regular Semigroups ◽

Completely Regular ◽

Form Part ◽

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.

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

Varieties of completely regular semigroups

10.1017/s144678870002365x ◽

1985 ◽

Vol 38 (3) ◽

pp. 372-393 ◽

Cited By ~ 23

Author(s):

Norman R. Reilly

Keyword(s):

Subdirect Product ◽

Variety Of Algebras ◽

Regular Semigroups ◽

Basis Of Identities ◽

Completely Regular ◽

Lattice Of Subvarieties ◽

And Inversion ◽

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.

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

International Journal of Algebra and Computation ◽

10.1142/s0218196719500541 ◽

2019 ◽

Vol 29 (08) ◽

pp. 1383-1407 ◽

Cited By ~ 1

Author(s):

Jiří Kad’ourek

Keyword(s):

Lattice Of Varieties ◽

Regular Semigroups ◽

Completely Regular ◽

Kernel Class ◽

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].

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

On free products of completely regular semigroups

10.1017/s1446788700034844 ◽

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 ◽

Products Of Groups ◽

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.

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

On the lattice of varieties of completely regular semigroups

10.1017/s1446788700025726 ◽

1983 ◽

Vol 35 (2) ◽

pp. 227-235 ◽

Cited By ~ 12

Author(s):

P. R. Jones

Keyword(s):

Lattice Of Varieties ◽

Regular Semigroups ◽

Completely Regular ◽

Variety Of Groups ◽

Subdirect Products

AbstractSeveral morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).