A topological semigroup S is a
Hausdorff space together with a continuous associative multiplication on S. A
semigroup S is said to be uniquely divisible if each element of S has unique roots of
each positive integral order in S. The present paper concerns uniquely divisible
semigroups on the two-cell. The main result of this paper is a statement of equivalent
conditions for a commutative uniquely divisible semigroup on the two-cell to be the
continuous homomorphic image of the cartesian product of two threads. This
result is applied to determine the structure of commutative uniquely divisible
semigroups on the two-cell whose idempotent set consists of a zero and an
identity.
