  As an appropriate generalisation of the features of the classical (Schein)
theory of representations of inverse semigroups in $\mathscr{I}_{X}$, a theory
of representations of inverse semigroups by homomorphisms into complete
atomistic inverse algebras is developed. This class of 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.
