On semigroup amalgams

In the year 2013, Kristiina Rahkema wrote her Bachelor’s thesis regarding semigroup amalgams. Semigroup amalgams occur when two semigroups S_1 and S_2 intersect and form a subsemigroup U. In that case, we are interested whether we can find a semigroup W such that it contains S_1 and S_2 and where the intersection stays the same. When this occurs we say that the amalgam is embeddable. While amalgams are always embeddable in the case of groups, then it is not always true for semigroups. In the last part of her thesis, Kristiina provided some results about the embeddability and non-embeddability of some types of semigroup amalgams and in this presentation we will be providing these results with proofs.