On quasi completely regular semirings

We extend the concepts of a completely π-regular semigroup and a GV semigroup to semirings and find a semiring analogue of a structure theorem on GV semigroups. We also show that a semiring S is quasi completely regular if and only if S is an idempotent semiring of quasi skew-rings.     

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.     

On varieties of completely regular semigroups III

Communicated by J.M. Howie     

Congruences on quasi completely regular semirings

Semigroup Forum - We study quasi completely regular semirings through their congruence structures.     

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.     

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...     

Some relations on completely regular semigroups

A semigroup S is called a Clifford semigroup if it is completely regular and inverse. In this paper, some relations related to the least Clifford semigroup congruences on completely regular semigroups are characterized. We give the relation between Y and ξ on completely regular semigroups and get that Y ^* is contained in the least Clifford congruence on completely regular semigroups generally. Further, we consider the relation Y ^*, Y, ν and ε on completely simple semigroups and completely regular semigroups. This work is supported by Leading Academic Discipline Project of Shanghai Normal University, Project Number: DZL803 and General Scientific Research Project of Shanghai Normal University, No. SK200707.     

New concepts for completely regular semigroups

Let 𝒞? denotes the variety of completely regular semigroups considered with the unary operation of inversion. The global study of the lattice of subvarieties of 𝒞? depends heavily on various decompositions. Some of the most fruitful among these are induced by the kernel and the trace relations. In their turn, these relations are induced by the kernel and the trace relations on the lattice of congruences on regular semigroups. These latter admit the concepts of kernel and trace of a congruence. The kernel and the trace relations for congruences were transferred to kernel and trace relations on varieties but the kernel and trace got no analogue for varieties.

We supply here the kernel and the trace of a variety which induce the relations of their namesake. For the local and core relations, we also define the local and core of a variety. All the new concepts are certain subclasses of 𝒞?. In this way, we achieve considerable similarity of the new concepts with those for congruences. We also correct errors in two published papers.     

On free spectra of finite completely regular semigroups and monoids

We show that a finite completely regular semigroup has a sub-log-exponential free spectrum if and only if it is locally orthodox and has nilpotent subgroups. As a corollary, it follows that the Seif Conjecture holds true for completely regular monoids. In the process, we derive solutions of word problems of free objects in a sequence of varieties of locally orthodox completely regular semigroups from solutions of word problems in relatively free bands.     

Weakly J-covered completely regular semigroups

Abstract. 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.     

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 \({\mathcal{L(CR)}}\).