Some Semigroups on the Two-cell

A topological semigroup is a nonvoid, Hausdorff space, 5, together with a continuous, associative multiplication defined on 5. Here the term semigroup will always denote a topological semigroup. In [4], Wallace and Koch have shown that if the circle, B, is a semigroup such that B2 = B, then either B is a group, or the multiplication on B is of the trivial kind xy = x, or xy = y. In [6], Mostert and Shields give a description of a semigroup on the two-cell where the boundary of the two-cell relative to the plane is a group. The main results of this paper will be a description of a semigroup on the two-cell where the multiplication satisfies the following two conditions: one; for x and y in the boundary of the two-cell relative to the plane xy = x, and two; there is a zero but no other idempotent in the interior of the two-cell. The following theorem will be proved: