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.     

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.     

Weakly commutative Lie semigroups

Communicated by Dennison R. Brown     

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.     

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.     

Weakly separative weakly commutative semigroups

A semigroup S is called a weakly commutative semigroup if, for every a,b∈S, there is a positive integer n such that (ab) ⁿ ∈Sa∩bS. A semigroup S is called archimedean if, for every a,b∈S, there are positive integers m and n such that a ⁿ ∈SbS and b ^m ∈SaS. It is known that every weakly commutative semigroup is a semilattice of weakly commutative archimedean semigroups. A semigroup S is called a weakly separative semigroup if, for every a,b∈S, the assumption a ²=ab=b ² implies a=b. In this paper we show that a weakly commutative semigroup is weakly separative if and only if its archimedean components are weakly cancellative. This result is a generalization of Theorem 4.16 of Clifford and Preston (The Algebraic Theory of Semigroups, Am. Math. Soc., Providence, 1961).