A major result in semigroup theory says that certain algebraic semigroups are isomorphic to semigroups of rectangular matrices, provided the matrices are multiplied using a fixed 'sandwich matrix' : that is, , for all matrices and in the semigroup.

In 1967, Magill defined a semigroup of transformations in a similar way: namely, suppose and are sets, let be the set of all mappings from into , fix some in , and define on by: . This produces a semigroup, and Magill determined when it is regular. Since then, some Thai mathematicians and I have extended this idea to partial transformations of arbitrary sets and vector spaces, and determined when such semigroups satisfy other algebraic properties. I will discuss some of that work in this talk.