A semigroup S is said
to be uniquely representable in terms of two subsets X and Y of S if
X ⋅ Y = Y ⋅ X = S,x1y1= x2y2 is a nonzero element of S implies x1= x2 and
y1= y2, and ylXl = y2x2 is a nonzero element of S implies y1= y2 and x1= x2 for
x1,x2∈ X and y1,y2∈ Y. A semigroup S is said to be uniquely divisible if for each
s ∈ S and every positive integer n there exists a unique z ∈ S such that zn= s.
Theorem. If S is a uniquely divisible semigroup on the two-cell with the set of
idempotents of S being a zero for S and an identity for S, then S is uniquely
representable in terms of X and Y where X and Y are iseomorphic copies of the
usual unit interval and the boundary of S equals X union Y . Corollary. If S is a
uniquely divisible semigroup on the two-cell and if S has only two idempotents, a
zero and an identity, then the nonzero elements of S form a cancellative
semigroup.
