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 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.

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.

REPRESENTING SUBDIRECT PRODUCT MONOIDS BY GRAPHS

2009 ◽  
Vol 19 (05) ◽  
pp. 705-721 ◽  
Author(s):  
VÁCLAV KOUBEK ◽  
VOJTĚCH RÖDL ◽  
BENJAMIN SHEMMER
Keyword(s):  
Subdirect Product ◽  
Abelian Groups ◽  
Endomorphism Monoid ◽  
Completely Simple Semigroups ◽  
Finite Monoids ◽  
Minimum Number ◽  
Completely Simple

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In a previous paper, we give a construction G(M) for a graph with prescribed endomorphism monoid M known as a [Formula: see text]-graph. Using this construction, we derived bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. In this paper, we generalize the [Formula: see text]-graph construction and derive several new bounds for monoid classes not handled by our first paper. Among these are the so called "strong semilattices of C-semigroups" where C is one of the following: Groups, Abelian Groups, Rectangular Groups, and completely simple semigroups.

scholarly journals Varieties of groups and of completely simple semigroups

1981 ◽  
Vol 23 (3) ◽  
pp. 339-359 ◽  
Author(s):  
Mario Petrich ◽  
Norman R. Reilly
Keyword(s):  
Subdirect Product ◽  
Completely Simple Semigroups ◽  
Completely Simple

Completely simple semigroups form a variety if we consider them both with the multiplication and the operation of inversion. Denote the lattice of all varieties of completely simple semi-groups by L(CS) and that of varieties of groups by L(G). We prove that the mappings V → V ∩ G and V → V v G are homomorphisms of L(CS) onto L(G) and the interval [G, CS], respectively. The homomorphism V → (V ∩ G, V v G) is an isomorphism of L(CS) onto a subdirect product. We explore different properties of the congruences on L(CS) induced by these homomorphisms.

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.

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 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.

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