ABSTRACT
For reasons similar to those of the group algebra case, see [203], one may define the classes of u.p. and t.u.p. semigroups. Namely, S is said to be a u.p. (unique-product) semigroup if, for any nonempty finite subsets X, Y of S with |X| + |Y| > 2, there exists an element in the set XY = {xy |x ∈ X, y ∈ Y] that has a unique presentation in the form xy, where x ∈ X,y ∈ Y. Similarly, S is called a t.u.p. (two-unique-product) semigroup if, for any nonempty finite subsets X, Y with |X| + |Y| > 2, there exist at least two elements in XY that have unique presentations as xy, for some x ∈ X, y ∈ Y. Clearly, every t.u.p. semigroup is a u.p. semigroup. Groups with the above-defined properties are called u.p. and t.u.p. groups, respectively.
