ON MAXIMAL SUBGROUPS OF FREE OBJECTS OF CERTAIN COMPLETELY REGULAR SEMIGROUP VARIETIES

2011 ◽  
Vol 21 (03) ◽  
pp. 473-484
Author(s):  
IGOR DOLINKA
Keyword(s):  
Regular Semigroup ◽  
Semigroup Variety ◽  
Maximal Subgroups ◽  
Completely Regular Semigroup ◽  
Locally Finite ◽  
Completely Regular ◽  
Free Generators ◽  
Free Semigroups ◽  
Semigroup Varieties ◽  
Relatively Free Group

By adjusting a method of Kadourek and Polák developed for free semigroups satisfying xr ≏ x, we prove that if [Formula: see text] is a periodic group variety, then any maximal subgroup of the free object in the completely regular semigroup variety of the form [Formula: see text] is a relatively free group in [Formula: see text] over a suitable set of free generators. When [Formula: see text] is locally finite, we provide some bounds for the sizes of its finitely generated members.

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.

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)

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