We describe a method for associating some non-self-adjoint algebras to MauldinWilliams graphs and we study the Morita equivalence and isomorphism of these algebras. We also investigate the relationship between the Morita equivalence and isomorphism class of the C∗-correspondences associated with Mauldin-Williams graphs and the dynamical properties of the Mauldin-Williams graphs.

Since 2005 a new powerful invariant of an algebra has emerged using the earlier work of Horvath, Hethelyi, Kulshammer and Murray. The authors studied Morita invariance of a sequence of ideals of the center of a nite dimensional algebra over a eld of nite characteristic. It was shown that the sequence of ideals is actually a derived invariant, and most recently a slightly modied version o...

The classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism infinite rings. In our main result, we establish analogous statements graded homomorphisms,

We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson’s concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent universal groupoids in the sense of Paterson and hence strongly Morita equivalent universal and reduced C∗-algebras. As a consequence we obtain a new proof of a result of Khoshkam and Skandalis showing t...