Reuben W. Farley

Copyright © 1985 Reuben W. Farley. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A positive semiroup is a topological semigroup containing a subsemigroup N isomorphic to the multiplicative semigroup of nonnegative real numbers, embedded as a closed subset of E2 in such a way that 1 is an identity and 0 is a zero. Using results in Farley [1] it can be shown that positive commutative semigroups on the plane constructed by the techniques given in Farley [2] cannot contain an infinite number of two dimensional groups. In this work an example of such a semigroup will be constructed which does, however, contain an infinite number of one dimensional groups. Also, some preliminary results are given here concerning what types of semilattices of idempotent elements are possible for positive commutative semigroups on E2. In particular, we will show that there is a unique positive commutative semigroup on E2 which is the union of connected groups and which contains five idempotent elements. Also, we will show that such semigroups having nine idempotent elements are not unique by constructing an example of such a semigroup with nine idempotent elements whose semilattice of idempotent elements is not “symmetric” and hence which is not isomorphic to the semigroup with nine idempotent elements constructed in Farley [2].