Representations of inverse semigroups in complete atomistic inverse meet-semigroups

As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in I-X, a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse perpendicular to-semigroups is developed. This class of inverse perpendicular to-semigroups, otherwise known as inverse algebras, includes partial automorphism monoids of entities such as graphs, vector spaces and modules. A workable theory of decompositions is reached; however complete distributivity is required for results approaching those of the classical case.