The pseudovariety generated by completely 0-simple semigroups

A finite basis of pseudoidentities of the pseudovariety generated by all finite completely 0-simple semigroups is constructed. Thus this pseudovariety is decidable. Partially supported by Israel Ministry of Absorption     

Varieties of structurally trivial semigroups II

Melnik [5] determined completely the lattice of all 2-nilpotent extensions of rectangular band varieties; and Koselev [4] determined a distributive sublattice formed by certain varieties of n-nilpotent extensions of left zero bands. In [2] the author described the skeleton of the lattice of all 3-nilpotent extensions of rectangular bands. We generalize these results by proving that a certain family of semigroup varieties which includes all the varieties mentioned above, and referred to here as planar varieties, consisting of certain n-nilpotent extensions of rectangular bands forms a distributive sublattice that looks somewhat like an inverted pyramid. Our proof makes use of a countably infinite family of injective endomorphisms on the lattice of all semigroup varieties that was introduced by the author in [1]. Although we do not determine completely the lattice of all n-nilpotent extensions of rectangular band varieties, our result unifies certain previously known results and provides a framework for further research.     

Varieties of structurally trivial semigroups I

Some useful recursive relations involving certain varieties of structurally trivial semigroups are proved. Using these relations, the skeleton of the lattice of all varieties of 3-nilpotent extensions of rectangular bands is shown to look somewhat like an inverted pyramid. In a subsequent paper [4] these results are generalised further to the variety of all n-nilpotent extensions of rectangular bands.     

Demidirectly closed pseudovarieties of locally trivial semigroups

Communicated by N.R. Reilly     

The join of the pseudovarieties of idempotent semigroups and locally trivial semigroups

This article solves a problem proposed by Almeida: the computation of the join of two well-known pseudovarieties of semigroups, namely the pseudovariety of bands and the pseudovariety of locally trivial semigroups. We use a method developed by Almeida, based on the theory of implicit operations. This work was partly supported by PRC Mathématiques et Informatique and by ESPRITBRA WG 6317 ASMICS-2     

The joint of the pseudovarieties of idempotent semigroups and locally trivial semigroups

Communicated by J.-E. Pin     

A class of semigroups having almost trivial multiplications

In this note we completely describe the structure of algebraic semigroups S with Sx=S or Sx degenerate and xS=S or xS degenerate for each x∈S. We then apply our results in characterizing the separately continuous multiplications on a topological space whose only self-maps are either surjective or have degenerate image. In particular, we find that any locally compact Hausdorff space with this property can admit only trivial separately continuous multiplications. Examples of spaces satisfying this property are certain continua discovered by H. Cook [1]. The authors are indebted to Karl H. Hofmann and James T. Rogers for conversations elucidating the topological implications of Theorem 1. The second author supported by NSF Grant GP28655     

The identities of the free product of two trivial semigroups

We exhibit an example of a finitely presented semigroup S with a minimum number of relations such that the identities of S have a finite basis while the monoid obtained by adjoining 1 to S admits no finite basis for its identities. Our example is the free product of two trivial semigroups.     

Some properties of E-m semigroups

In this note we study some properties of E-m semigroups, recently defined by Nordahl in [3]. In particular we study connections with power joined and strongly reversible semigroups. We prove, among other things, that an E-m semigroup is a disjoint union of power joined semigroups.The second part of the paper deals with case m = 2, which presents interesting properties. For undefined terminology and notation the reader is referred to [2].     

Some subdirectly irreducible idempotent semigroups

Subdirectly irreducible idempotent semigroups were characterized in [3], and in that paper, their connection with the various equational classes of idempotent semigroups was discussed. All these results are in terms of identities, so that examples of subdirectly irreducibles in the equational classes are explicitly known only for small classes. It is easy to show from general considerations (see the last section of the present paper) that every proper equational subclass of the class of idempotent semigroups is generated (as an equational class) by one or two subdirectly irreducibles. In this paper we give an example of a subdirectly irreducible for each join irreducible equational class of idempotent semigroups, which generates the class. This list, together with known results, gives explicit examples of one or two finite subdirectly irreducibles which generate the various equational classes. Research supported by the National Research Council of Canada.     

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.