On varieties of completely regular semigroups III

Communicated by J.M. Howie     

On some intervals of varieties of completely regular semigroups

Semigroup Forum - Completely regular semigroups with the unary operation of inversion within their maximal subgroups form a variety under inclusion denoted by $$mathcal {C}mathcal {R}$$. The...     

Clusters of varieties of completely regular semigroups

The class \({\mathcal{CR}}\) of completely regular semigroups equipped with the unary operation of inversion forms a variety whose lattice of subvarieties is denoted by \({\mathcal{L(CR)}}\). The variety \({\mathcal B}\) of all bands induces two relations \({\mathbf{B}^{\land}}\) and \({\mathbf{B}^{\lor} }\) by meet and join with \({\mathcal B}\). Their classes are intervals with lower ends \({\mathcal V_{B^{\land}}}\) and \({\mathcal V_{B^{\lor}}}\), and upper ends \({\mathcal V^{B^{\land}}}\) and \({\mathcal V^{B^{\lor}}}\). These objects induce four operators on \({\mathcal{L(CR)}}\).The cluster at a variety \({\mathcal V}\) is the set of all varieties obtained from \({\mathcal V}\) by repeated application of these four operators. We identify the cluster at any variety in \({\mathcal{L(CR)}}\).     

Another lattice of varieties of completely regular semigroups

Completely regular semigroups S are taken here with the unary operation of inversion within the maximal subgroups of S. As such they form a variety 𝒞? whose lattice of subvarieties is denoted by ?(𝒞?). The relation on ?(𝒞?) which identifies two varieties if they contain the same bands is denoted by B^∧. The upper ends of B^∧-classes which are neither equal to 𝒞? nor contained in the variety 𝒞𝒮 of completely simple semigroups are generated by two countably infinite ascending chains called canonical varieties. In a previous publication, we constructed the sublattice Σ of ?(𝒞?) generated by 𝒞𝒮 and the first four canonical varieties. Here we extend Σ to the sublattice Ψ of ?(𝒞?) generated by 𝒞𝒮 and the first six canonical varieties. For each of the varieties in Ψ?Σ, we construct the ladder and a basis of its identities.     

Ladders and canonical varieties of completely regular semigroups

A completely regular semigroup is a (disjoint) union of its (maximal) subgroups. We consider it here with the unary operation of inversion within its maximal subgroups. Their totality \(\mathcal {C}\mathcal {R}\) forms a variety whose lattice of subvarieties is denoted by \(\mathcal {L}(\mathcal {C}\mathcal {R})\). On it, one defines the relations \(\mathbf {B}^\wedge \) and \(\mathbf {B}^\vee \) by

$$\begin{aligned} \begin{array}{lll} \mathcal {U}\ \mathbf {B}^\wedge \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\cap \mathcal {B} =\mathcal {V}\cap \mathcal {B}, \\ \mathcal {U}\ \mathbf {B}^\vee \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\vee \mathcal {B} =\mathcal {V}\vee \mathcal {B} , \end{array} \end{aligned}$$

respectively, where \(\mathcal {B}\) denotes the variety of all bands. This is a study of the interplay between the \(\cap \)-subsemilatice \(\triangle \) of \(\mathcal {L}(\mathcal {C}\mathcal {R})\) of upper ends of \(\mathbf {B}^\wedge \)-classes and their \(\mathbf {B}^\vee \)-classes. The main tool is the concept of a ladder and their \(\mathbf {B}^\vee \)-classes, an indispensable part of the important Polák’s theorem providing a construction for the join of varieties of completely regular semigroups. The paper includes the tables of ladders of the upper ends of most \(\mathbf {B}^\wedge \)-classes. Canonical varieties consist of two ascending countably infinite chains which generate most of the upper ends of \(\mathbf {B}^\wedge \)-classes.

     

On some presentations of completely regular semigroups

In this paper we consider three types of presentations of completely regular semigroups. In each of the considered cases the solution of the word problem can be reduced to the solution of the word problem for a corresponding group presentation. As a consequence, in each of these cases the one relator presentation has a solvable word problem.     

Weakly J-covered completely regular semigroups

After defining the weakly covering and covering congruence on regular semigroups, we give a necessary and sufficent condition for the J-relation on a completely regular semigroup to be a weakly covering congruence and construct J-covered and weakly covered completely regular semigroups.