Vol. 21, No. 2, 1967

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On compact unithetic semigroups

John A. Hildebrant

Vol. 21 (1967), No. 2, 265–273

A topological semigroup is a Hausdorff space ∕S together with a continuous, associative multiplication. If each element of S has unique roots in S of each positive integral order, then S is said to be uniquely divisible. The closure of the set of positive rational powers of an element x in a compact uniquely divisible semigroup S is a commutative clan (compact connected semigroup with identity) called the unithetic semigroup generated by x.

The purpose of this paper is to discuss the structure of compact unithetic semigroups. It is established that if the cartesian product of two semigroups is unithetic, then both factors are unithetic, and at least one factor is a group.

A partial converse is presented. If S is a compact first countable unithetic semigroup, and G is a finite dimensional compact unithetlc group, then G × S is a unithetic semigroup. These results are used to give the precise of a unithetic semigroup with zero whose maximal group containing the identity is finite dimensional. A complete converse to the first result is not known. In particular, the question as to whether one or both of the conditions that S be first countable and G be finite dimensional can be omitted is open.

Mathematical Subject Classification
Primary: 22.05
Received: 13 April 1966
Published: 1 May 1967
John A. Hildebrant