Uniquely representable semigroups. II
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 if X-Y -Y-X = S 9 x x y u -#22/2 is a nonzero element of S implies Xi = x 2 and y x = y 2 and ViXi = VzX2 is a nonzero element of S implies y γ = y 2 and Xι = x 2 for all x u x 2 eX and y u y 2 e Y.In this paper we are concerned with semigroups S with no zero divisors, E(S) = {0,1}, and which are uniquely representable in terms of two subsets X and Y which are iseomorphic copies of the unusual unit interval.Here we show the nonzero elements of the semigroup S can be embedded in a Lie group.The authors would like to express their appreciation to Professor G. D. Crown for taking part in discussions concerning this paper.NOTATION.S will represent a semigroup without zero-divisors, E(S) = {0,1} (E(S) is the set of idempotents of S), and which is uniquely representable in terms of X and Y which are isomorphic copies of the usual unit interval.We will let T = S -{0} where 0 is the zero of S. Also X° = X -{0} and X 01 -X -{0,1} where 1 is the identity for S. Similarly, Y° = Y -{0}, and Y 01 = Y -{0,1}.Define φ: Γx Γ^Γx Y° by φ(x, y) = (x f , y') where x' and y' are the unique elements of X° and Y° respectively such that xy -y'x'.Also define ψ: Γx Y°-> X° x Y° by ψ(x, y) = (x', y') where x f and y' are the unique elements of X° and Y° respectively such that yx = χ r