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.     

On left cancellative semigroups

In this note we shall describe the structure of left cancellative semigroups. We shall also investigate the existence of idempotents and show that the existence of idempotents is related to the existence of maximal proper right ideals.     

Commutativity theorems for cancellative semigroups

Psomopoulos has proved that \([x^n, y] = [x, y^{n+1}]\) for a positive integer n implies commutativity in groups. Here we show that cancellative semigroups admitting commutators and satisfying the identity \([x^n, y] = [x, y^{n+k}]\) implies that the element \(y^k\) is central. The special case of \(k=1\) yields the above mentioned commutativity theorem. To accommodate negative exponents, we consider the functional equation \([f(x), y] = [x, g(y)f(y)] \) where f and g are unary functions satisfying certain formal syntactic rules and prove that in cancellative semigroups admitting commutators, the functional equation \([f(x), y] = [x, g(y)f(y)]\) implies that the element g(y) is central i.e. \(xg(y) = g(y)x\) for all x and y. By the way, these results are new even in group theory.     

The rank of a commutative cancellative semigroup

Summary For a commutative cancellative semigroup S, we define the rank of S intrinsically. This definition implies that the rank of S equals the usual rank of its group of quotients. We also characterize the rank in terms of embeddability into a rational vector space of the greatest power cancellative image of S.     

Commutative cancellative semigroups of finite rank

The rank of a commutative cancellative semigroup S is the cardinality of a maximal independent subset of S. Commutative cancellative semigroups of finite rank are subarchimedean and thus admit a Tamura-like representation. We characterize these semigroups in several ways and provide structure theorems in terms of a construction akin to the one devised by T. Tamura for N-semigroups.     

More on cancellative semigroups on manifolds

Communicated by D.R.Brown     

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.     

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     

On finitely presented,cancellative and commutative ordered monoids

We show that every finitely presented, cancellative and commutative ordered monoid is determined by a finitely generated and cancellative pseudoorder on the monoid (ℕ ⁿ ,+) for some positive integer n. Every cancellative pseudoorder on (ℕ ⁿ ,+) is determined by a submonoid of the group (ℤ ⁿ ,+), and we prove that the pseudoorder is finitely generated if and only if the submonoid is an affine monoid in ℤ ⁿ .     

Prime and Semiprime Semigroup algebras of cancellative semigroups

Necessary and sufficient conditions are given for a semigroup algebra of a cancellative semigroup to be prime and semiprime. These conditions were proved necessary by Okninski; our contribution is to show that they are also sufficient. The techniques used in the proof are a new variation on the -methods which were developed originally for group algebras.

     

真理想为Archimedean子半群的偏序半群

根据真理想情况给出了偏序半群的一种分类，研究了真理想为Archimedean子半群的偏序半群的特征．