Abstract
A Morita equivalence similar to that found by Green for crossed products by groups will be established for crossed products by inverse semigroups. More precisely, let S be an inverse semigroup, T a finite sub-inverse semigroup of S and A an S-algebra or a T-algebra. Then the crossed product $$A \rtimes T$$ is Morita equivalent to a certain crossed product $$B \rtimes S$$.
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