E-varieties of regular semigroups,relatively bifree objects and fully invariant congruences

Analogous to the concept of a free object on a setX in a variety of algebras is the notion of a bifree object onX in an e-variety of regular semigroups. If an e-variety contains a bifree object onX, then a homomorphic image of that bifree object is itself bifree onX in some e-variety if and only if the corresponding congruence is fully invariant. Furthermore, the lattice of e-subvarieties of any locally inverse or E-solid e-variety ε is antiisomorphic with the lattice of all fully invariant congruences on the bifree object on a countably infinite setX in ε. We give a Birkhoff-type theorem for classes of locally inverse or E-solid semigroups, and we give an intrinsic test for whether or not a regular semigroup is bifree onX in the e-variety it generates.     

The chain of right simple semigroups in ordered semigroups

The relation ≤ is defined on the set of right ideals of an ordered semigroup. The main result of this paper is as follows: an ordered semigroup S is a chain of right simple ordered semigroups if and only if ≤ is an order relation. Bibliography: 3 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 83–88.     

The structure ofP-regular semigroups

This paper gives some equivalent definitions of stronglyP-regular semigroups and characterizes the structure ofP-regular semigroups as the spined product of fundamentalP-regular semigroups and regular *-semigroups. This work is supported by the National Nature Science Foundation of China.     

Central idempotents of the bifree and left-free double Burnside ring

We determine the blocks, i.e., the primitive central idempotents, of the bifree double Burnside ring and the left-free double Burnside ring, as well as the primitive central idempotents of the algebras arising from scalar extension to ?.     

The modulus of matrix semigroups

We treat a result concerning the generator of the modulus semigroup of a strongly continuous semigroup acting on the product of Banach lattices with order continuous norm. Received: 28 March 2003     

The structure of superabundant semigroups

A structure theorem for superabundant semigroups in terms of semilattices of normalized Rees matrix semigroups over some cancellative monoids is obtained. This result not only provides a construction method for superabundant semigroups but also generalizes the well-known result of Petrich on completely regular semigroups. Some results obtained by Fountain on abundant semigroups are also extended and strengthened.     

The generators of positive semigroups

We characterise the infinitesimal generators of norm continuous one-parameter semigroups of positive maps on certain ordered spaces, with special reference to C¹-algebras.     

Quotient semigroups and extension semigroups

We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of C ^*-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.     

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     

Invariant semigroups of orthodox semigroups

We consider, in a right inverse semigroupS with a multiplicative inverse transversalS ^o, the notion of anS ^o-invariant subsemigroup and use this to describe all the left amenable orders definable onS. The results obtained, together with their duals, are used to prove that ifS is an orthodox semigroup with a multiplicative inverse transversalS ^o, then every amenable order onS ^o can be extended to a unique amenable order onS. NATO Collaborative Research Grant 910765 is gratefully acknowledged. The second-named author also gratefully acknowledges support from the Calouste Gulbenkian Foundation, Lisbon.     

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

Communicated by J.-E. Pin