Uniquely representable semigroups on the two-cell
J. T. Borrego, Haskell Cohen, E. E. DeVun · Pacific Journal of Mathematics · 1971
A semigroup S is said to be uniquely representable in terms of two subsets X and Y of S if XΎ=Y-X=S, x^i = x 2 y 2 is a nonzero element of S implies x x -x 2 and y x = y 2f and ViXi = 2/2^2 is a nonzero element of S implies y^ = y 2 and Xι -x 2 for x lt x 2 eX and y ly y 2 eY.A semigroup S is said to be uniquely divisible if for each s e S and every positive integer n there exists a unique zeS such that z n = 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 £ 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.