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.

Certain Varieties and Quasivarieties of Completely Regular Semigroups

1977 ◽  
Vol 29 (6) ◽  
pp. 1171-1197 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Unique Element ◽  
Completely Regular Semigroup ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups ◽  
Definition Of ◽  
Image Position

We adopt the following definition of a completely regular semigroup S: for every element a of S, there exists a unique element a-1 of S such that

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).

The Word Problem for Orthogroups

1981 ◽  
Vol 33 (4) ◽  
pp. 893-900 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Maximal Subgroup ◽  
Word Problem ◽  
Unary Operation ◽  
Group Inverse ◽  
Completely Regular Semigroup ◽  
Completely Regular ◽  
Image Position ◽  
Natural Way

A semigroup which is a union of groups is said to be completely regular. If in addition the idempotents form a subsemigroup, the semigroup is said to be orthodox and is called an orthogroup. A completely regular semigroup S is provided in a natural way with a unary operation of inverse by letting a-l for a ∈ S be the group inverse of a in the maximal subgroup of S to which a belongs. This unary operation satisfies the identities(1)(2)(3)In fact a completely regular semigroup can be defined as a unary semigroup (a semigroup with an added unary operation) satisfying these identities. An orthogroup can be characterized as a completely regular semigroup satisfying the additional identity(4)

scholarly journals Free completely regular semigroups

1984 ◽  
Vol 25 (2) ◽  
pp. 241-254 ◽  
Author(s):  
P. G. Trotter
Keyword(s):  
Regular Semigroup ◽  
Structure Theory ◽  
Completely Regular Semigroup ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Additional Operation ◽  
Alternative Characterization ◽  
Completely Regular Semigroups

A completely regular semigroup is a semigroup that is a union of groups. The aim here is to provide an alternative characterization of the free completely regular semigroup Fcrx on a set X to that given by J. A. Gerhard in [3, 4].Although the structure theory for completely regular semigroups was initiated in 1941 [1] by A. H. Clifford it was not until 1968 that it was shown by D. B. McAlister [5] that Fcrx exists. More recently, in [7], M. Petrich demonstrated the existence of Fcrx by showing that completely regular semigroups form a variety of unary semigroups (that is, semigroups with the additional operation of inversion).

scholarly journals The lattice of completely regular semigroup varieties

1990 ◽  
Vol 49 (1) ◽  
pp. 24-42 ◽  
Author(s):  
Francis Pastijn
Keyword(s):  
Regular Semigroup ◽  
Completely Regular Semigroup ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Subdirect Decomposition ◽  
Closure Operations ◽  
Completely Regular Semigroups ◽  
Semigroup Varieties

AbstractA completely regular semigroup is a semigroup which is a union of groups. The class CR of completely regular semigroups forms a variety. On the lattice L (CR) of completely regular semigroup varieties we define two closure operations which induce complete congruences. The consideration of a third complete congruence on L (CR) yields a subdirect decomposition of L (CR). Using these results we show that L (CR) is arguesian. This confirms the (tacit) conjecture that L (CR) is modular.

Free generators in relatively free completely regular semigroups

1988 ◽  
Vol 109 (3-4) ◽  
pp. 329-339
Author(s):  
P.G. Trotter
Keyword(s):  
Regular Semigroup ◽  
Completely Regular Semigroup ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Free Objects ◽  
Free Generators ◽  
Completely Regular Semigroups ◽  
Similar Description ◽  
Free Semilattice

SynopsisA subset Y of a free completely regular semigroup FCRx freely generates a free completely regular subsemigroup if and only if (i) each -class of FCRx contains at most one element of Y, (ii) {Dy;y ∊ Y} freely generates a free subsemilattice of the free semilattice FCRx/), and (iii) Y consists of non-idempotents. A similar description applies in free objects of some subvarieties of the variety of all completely regular semigroups.

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 Operators related to idempotent generated and monoid completely regular semigroups

1990 ◽  
Vol 49 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Mario Petrich ◽  
Norman R. Reilly
Keyword(s):  
Binary Operation ◽  
Unary Operation ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups

AbstractThe class CR of completely regular semigroups (unions of groups or algebras with the associative binary operation of multiplication and the unary operation of inversion subject to the laws x = xx-1, (x−1)-1 = x and xx-1 = x-1x) is a variety. Among the important subclasses of CR are the classes M of monoids and I of idempotent generated members. For each C ∈ {I, M}, there are associated mappings ν → ν ∩ C and ν → (Ν ∩ C), the variety generated by ν ∩ C. The lattice theoretic properties of these mappings and the interactions between these mappings are studied.

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.

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