Amalgamating inverse semigroups over non-inverse semigroups

An amalgam (S; T1, T2) of inverse semigroups is long known to be embeddable in an inverse semigroup. Also, for any semigroups T1 and T2 the amalgam (S; T1, T2) is embeddable if S is an inverse semigroup. On the other hand, we know that (S; T1, T2) is non-embeddable if T1 and T2 are both groups but S is not. In my previous seminar, I discussed some examples showing that (S; T1, T2) may not be embeddable if T1 and T2 are both inverse semigroups but S is not.

In this seminar I will show that (S; T1, T2) is, however, weakly embeddable if T1 and T2 are both inverse semigroups while S is a non-inverse semigroup.