Vol. 38, No. 3, 1971

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ISSN: 0030-8730
Uniquely representable semigroups on the two-cell

J. T. Borrego, Haskell Cohen and Esmond Ernest Devun

Vol. 38 (1971), No. 3, 565–569
Abstract

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.

Mathematical Subject Classification 2000
Primary: 22A15
Milestones
Received: 10 December 1969
Published: 1 September 1971
Authors
J. T. Borrego
Haskell Cohen
Esmond Ernest Devun