Globalizations of strong partial acts over monoids

Let S be a monoid. Then a strong partial S-act is a partial S-act that arises by omitting some elements from a global S-act. If A is a partial act and B a globalization of A that is generated by the elements of A, then we say that B is an A-generated globalization of A. Kellendonk and Lawson have shown that if S is a group, then any strong partial S-act has a unique A-generated globalization. This however is not the case for monoids. Laan and Kudryavtseva gave two constructions for globalizing partial semigroup acts: the tensor product globalization \(A\otimes S\) $A\otimes S$ and the hom-set globalization \(A^S\) $A^S$ . They then showed that these constructions need not be isomorphic. In this paper we give a definition of the hom-set globalization on morphisms of partial acts which gives a faithful functor from the category of strong partial S-acts to global S-acts which is neither a reflector nor a coreflector. We show that isomorphism classes of A-generated globalizations form a complete lattice that is dual to a sublattice of \({\text {Con}}(A\otimes S)\) $\text{Con}(A\otimes S)$ . Lastly, we prove that groups are the only monoids for which all strong partial acts are uniquely globalizable.