Strong Morita Equivalence of Inverse Semigroups
نویسنده
چکیده
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 that the C∗-algebra of an F -inverse semigroup is strongly Morita equivalent to a cross product of a commutative C∗-algebra by a group.
منابع مشابه
Characterisations of Morita equivalent inverse semigroups
For a fixed inverse semigroup S, there are two natural categories of left actions of S: the category Fact of unitary actions of S on sets X meaning actions where SX = X, and the category Étale of étale actions meaning those unitary actions equipped with a function p : X → E(S), to the set of idempotents of S, such that p(x)x = x and p(sx) = ses∗, where s∗ denotes the inverse of s. The category ...
متن کاملMorita Equivalence of Semigroups with Local Units
We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locall...
متن کاملStrong Morita equivalence for semigroups with local units
Morita equivalence is a widely used tool for rings with identity. (Two rings are said to be Morita equivalent if the categories of unitary modules over them are equivalent.) For monoids, this notion is not really useful: in most cases it reduces to isomorphism. As the theory of Morita equivalence could be developed for the more general case of rings with local units, and then for idempotent rin...
متن کاملMorita invariants for partially ordered semigroups with local units
We study Morita invariants for strongly Morita equivalent partially ordered semigroups with several types of local units. These include the greatest commutative images, satisfying a given inequality and the fact that strong Morita equivalence preserves various sublattices of the lattice of ideals.
متن کاملMorita equivalence based on Morita context for arbitrary semigroups
In this paper, we study the Morita context for arbitrary semigroups. We prove that, for two semigroups S and T, if there exists a Morita context (S, T, P,Q, τ, μ) (not necessary unital) such that the maps τ and μ are surjective, the categories US-FAct and UT -FAct are equivalent. Using this result, we generalize Theorem 2 in [2] to arbitrary semigroups. Finally, we give a characterization of Mo...
متن کامل