Uniquely representable semigroups on the two-cell

Abstract

A semigroup S is said to be uniquely representable in terms of two subsets X and Y of S if X=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 X -x 2 for x lt x 2 eX and y ly y 2 eY.