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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.
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