On the word problem for free bands of groups and for free objects in some other varieties of completely regular semigroups

1989 ◽  
Vol 38 (1) ◽  
pp. 1-55 ◽  
Author(s):  
Jiří Kaďourek
Keyword(s):  
Word Problem ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Free Objects ◽  
Completely Regular Semigroups

scholarly journals The pseudoidentity problem and reducibility for completely regular semigroups

2001 ◽  
Vol 63 (3) ◽  
pp. 407-433 ◽  
Author(s):  
Jorge Almedia ◽  
Peter G. Trotter
Keyword(s):  
Word Problem ◽  
Finite Groups ◽  
Open Problem ◽  
Sufficient Conditions ◽  
90Th Birthday ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Strengthened Version ◽  
Completely Regular Semigroups ◽  
Necessary And Sufficient

Dedicated to George Szekeres on the occasion of his 90th birthdayNecessary and sufficient conditions for equality over the pseudovariety CR of all finite completely regular semigroups are obtained. They are inspired by the solution of the word problem for free completely regular semigroups and clarify the role played by groups in the structure of such semigroups. A strengthened version of Ash's inevitability theorem (κ-reducibility of the pseudovariety G of all finite groups) is proposed as an open problem and it is shown that, if this stronger version holds, then CR is also κ-reducible and, therefore, hyperdecidable.

scholarly journals Hyperdecidability of pseudovarieties of orthogroups

2001 ◽  
Vol 43 (1) ◽  
pp. 67-83 ◽  
Author(s):  
Jorge Almeida ◽  
Peter G. Trotter
Keyword(s):  
Word Problem ◽  
Weak Inversion ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups ◽  
Mal'cev Product ◽  
Mal’Cev Product

Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product of the pseudovariety of bands with a pseudovariety V of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that decidability is preserved in the case in which terms involving only multiplication and weak inversion are considered. It is also shown that, if V is a hyperdecidable (respectively canonically reducible) pseudovariety of groups, then so is W.

scholarly journals Word problems for free objects in certain varieties of completely regular semigroups

1983 ◽  
Vol 104 (2) ◽  
pp. 351-359 ◽  
Author(s):  
James Gerhard ◽  
Mario Petrich
Keyword(s):  
Word Problems ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Free Objects ◽  
Completely Regular Semigroups

scholarly journals Free completely regular semigroups II. Word problem

1983 ◽  
Vol 82 (1) ◽  
pp. 143-156 ◽  
Author(s):  
J.A Gerhard
Keyword(s):  
Word Problem ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups

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.

E-free objects and e-locality for completely regular semigroups

1995 ◽  
Vol 51 (1) ◽  
pp. 357-377 ◽  
Author(s):  
Peter R. Jones
Keyword(s):  
Regular Semigroups ◽  
Completely Regular ◽  
Free Objects ◽  
Completely Regular Semigroups

On the word problem for free completely regular semigroups

1986 ◽  
Vol 34 (1) ◽  
pp. 127-138 ◽  
Author(s):  
Jiří Kadourek ◽  
Libor Polák
Keyword(s):  
Word Problem ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups

scholarly journals Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox *-semigroup varieties

1993 ◽  
Vol 35 (1) ◽  
pp. 25-37 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Type Theorem ◽  
Orthodox Semigroup ◽  
Prima Facie ◽  
Direct Products ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Free Objects ◽  
Completely Regular Semigroups ◽  
Definition Of ◽  
Orthodox Semigroups

For regular semigroups, the appropriate analogue of the concept of a variety seems to be that of an e(xistence)-variety, developed by Hall [6,7,8]. A class V of regular semigroups is an e-variety if it is closed under taking direct products, regular subsemigroups and homomorphic images. For orthodox semigroups, this concept has been introduced under the term “bivariety” by Kaďourek and Szendrei [12]. Hall showed that the collection of all e-varieties of regular semigroups forms a complete lattice under inclusion. Further, he proved a Birkhoff-type theorem: each e-variety is determined by a set of identities. For e-varieties of orthodox semigroups a similar result has been proved by Kaďourek and Szendrei. At variance with the case of varieties, prima facie the free objects in general do not exist for e-varieties. For instance, there is no free regular or free orthodox semigroup. This seems to be true for most of the naturally appearing e-varieties (except for cases of e-varieties which coincide with varieties of unary semigroups such as the classes of all inverse and completely regular semigroups, respectively). This is true if the underlying concept of free objects is denned as usual. Kaďourek and Szendrei adopted the definition of a free object according to e-varieties of orthodox semigroups by taking into account generalized inverses in an appropriate way. They called such semigroups bifree objects. These semigroups satisfy the properties one intuitively expects from the “most general members” of a given class of semigroups. In particular, each semigroup in the given class is a homomorphic image of a bifree object, provided the bifree objects exist on sets of any cardinality. Concerning existence, Kaďourek and Szendrei were able to prove that in any class of orthodox semigroups which is closed under taking direct products and regular subsemigroups, all bifree objects exist and are unique up to isomorphism. Further, similar to the case of varieties, there is an order inverting bijection between the fully invariant congruences on the bifree orthodox semigroup on an infinite set and the e-varieties of orthodox semigroups. Recently, Y. T. Yeh [22] has shown that suitable analogues to free objects exist in an e-variety V of regular semigroups if and only if all members of V are either E-solid or locally inverse.

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).

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