Constructions of positive commutative semigroups on the plane. II (Q1060301)

A positive semigroup is a topological semigroup containing a subsemigroup N isomorphic to the multiplicative semigroup of nonnegative real numbers, embedded as a closed subset of \(E^ 2\) in such a way that 1 is an identity and 0 is a zero. Using results by the author [Trans. Am. Math. Soc. 151, 353-369 (1970; Zbl 0211.339)] it can be shown that positive commutative semigroups on the plane constructed by his techniques [Bull. Inst. Math. Acad. Sin. 7, 357-362 (1979; Zbl 0424.22004)] 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 \(E^ 2\). In particular, we will show that there is a unique positive commutative semigroup on \(E^ 2\) 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 by the author in part I.