CONSTRUCTIONS OF POSITIVE COMMUTATIVE SEMIGROUPS ON THE PLANE, II

ABSTRACT. A positive semiroup is a topological semigroup containinq a subseminroup N isomorphic to the multiplicative semigroup of nonnegative real numbers, embedded as a closed subset of E 2 in such a way that is an identity and 0 is a zero. Usina results in Farley [I] 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 semigrouD will be constructed which does, however, contain an infinite number of one dimensional groups. Also, some preliminary results are given here concerninQ what types of semilattices of idempotent elements are oossible for positive commutative semigroups on E 2. In particular, we will show that there is a unique positive commutative semi-group on E2 which is the union of connected groups and which contains five idempotent elements. Also, we will show that such semigroups havinq 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]. KEY WORDS AND PHRASES. Positive semigroup, Clifford semi.group, semiattioe of idempotent e ements