Weakly commutative Lie semigroups

Communicated by Dennison R. Brown     

Cancellative commutative semigroups

The study of t.c.r. (=totally cancellative reduced) real semigroups (which are just convex cones in real vector spaces, with the induced addition) enables us to answer some questions about t.c.r. semigroups in general. For example, a finite-dimensional divisible commutative semigroups is locally free if and only if it is t.c.r. and has the Riesz Interpolation Property. An address delivered at the Symposium on Semigroups and the multiplicative Structure of Rings in Mayagüez, 1970     

Pseudosimple commutative semigroups

As a simple corollary to the main result of [4] we describe the structure of commutative semigroups which are isomorphic to their nontrivial homomorphic images.     

Weakly B-orthodox semigroups

We introduce the notions of a band category and of a weakly orthodox category over a band. Our focus is to describe a class of weakly $B$ -orthodox semigroups, where $B$ denotes a band of idempotents. In particular, we investigate orthodox semigroups, by using orthodox groupoids. Weakly $B$ -orthodox semigroups are analogues of orthodox semigroups, where the relations $\widetilde{\mathcal {R}}_B$ and $\widetilde{\mathcal {L}}_B$ play the role that ${\mathcal {R}}$ and $\mathcal {L}$ take in the regular case. We show that the category of weakly $B$ -orthodox semigroups and admissible morphisms is equivalent to the category of weakly orthodox categories over bands and orthodox functors. The same class of weakly $B$ -orthodox semigroups was studied in an earlier article by Gould and the author using generalised categories. Our approach here is more akin to that of Nambooripad. The significant difference in strategy is that it is more convenient to consider categories equipped with pre-orders, rather than with partial orders.     

On power commutative semigroups

Communicated by G. Lallement     

Computing finite commutative semigroups

Precedence results are used to improve existing algorithms for the enumeration of finite commutative semigroups. As an application 11,545,843 distinct commutative semigroups of order 9 were found.     

On lattice-ordered commutative semigroups

Decompositions of elements into intersections of primal elements and into intersections of p-components are studied in certain lattice-ordered commutative semigroups, by making use of the new development in commutative ideal theory without finiteness conditions, due to Fuchs-Heinzer-Olberding [7]. Several results concerning ideals can be phrased as theorems in abstract ideal theory.The intersections we consider are in general not irredundant, and the associated prime elements are not unique. However, one can establish a canonical intersection that is often irredundant with uniquely determined associated primes.     

On weakly commutative peo-semigroups

Communicated by Robert McFadden     

Subdirectly irreducible right commutative semigroups

A semigroupS is said to be right commutative ifaxy=ayx for alla,x,y ∈ S. The object of this paper is to determine the subdirectly irreducible right commutative semigroups.