scholarly journals Subdirect decompositions of the lattice of varieties of completely regular semigroups

1989 ◽  
Vol 39 (3) ◽  
pp. 343-351 ◽  
Author(s):  
P.G. Trotter
Keyword(s):  
Complete Lattice ◽  
Principal Ideal ◽  
Subdirect Product ◽  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Distributive Element ◽  
Completely Regular ◽  
Completely Regular Semigroups

It is shown that if V is an element of the lattice of the title then the map given by U → (V ∧ U, V ∨ U) is a complete lattice embedding of into (V] × [V) if and only if V is a join-infinitely distributive element. In this case the image of the map is a subdirect product of the principal ideal (V] by the principal filter [V) generated by V. Some important varieties in are shown to be join-infinitely distributive.

scholarly journals On the lattice of varieties of completely regular semigroups

1983 ◽  
Vol 35 (2) ◽  
pp. 227-235 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Variety Of Groups ◽  
Completely Regular Semigroups ◽  
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).

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 Canonical Varieties of Completely Regular Semigroups

2007 ◽  
Vol 83 (1) ◽  
pp. 87-104 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Unary Operation ◽  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups

AbstractCompletely regular semigroups CR are regarded here as algebras with multiplication and the unary operation of inversion. Their lattice of varieties is denoted by L(CR). Let B denote the variety of bands and L(B) the lattice of its subvarieties. The mapping V → V ∩ B is a complete homomorphism of L(CR) onto L(B). The congruence induced by it has classes that are intervals, say VB = [VB, VB] for V ∈ L(CR). Here VB = V ∩ B. We characterize VB in several ways, the principal one being an inductive way of constructing bases for v-irreducible band varieties. We term the latter canonical. We perform a similar analysis for the intersection of these varieties with the varieties BG, OBG and B.

scholarly journals The modularity of the lattice of varieties of completely regular semigroups and related representations

1990 ◽  
Vol 32 (2) ◽  
pp. 137-152 ◽  
Author(s):  
Mario Petrich ◽  
Norman R. Reilly
Keyword(s):  
Regular Semigroup ◽  
Unary Operation ◽  
Direct Products ◽  
Completely Regular Semigroup ◽  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups ◽  
Subdirect Products ◽  
Image Position

A semigroup endowed with a unary operation satisfying the identitiesis a completely regular semigroup. In several recent papers devoted to the study of the lattice of subvarieties of the variety of completely regular semigroups, various results have been obtained which decompose special intervals in into either direct products or subdirect products. Petrich [14], Hall and Jones [6] and Rasin [20] have shown that certain intervals of the form , where is the trivial variety and are subdirect products of and Pastijn and Trotter [13] show that certain intervals of the form are direct products of the intervals and The main objective of this paper is to develop an appropriate lattice theoretic framework for these representations.

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].

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