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Two semigroups are called strongly Morita equivalent if they are contained in a Morita context with unitary bi-acts and surjective mappings. We consider the notion of context equivalence which is obtained from the notion of strong Morita equivalence by dropping the requirement of unitariness. We show that context equivalence is an equivalence relation on the class of factorisable semigroups and describe factorisable semigroups that are context equivalent to monoids or groups, and semigroups with weak local units that are context equivalent to inverse semigroups, orthodox semigroups or semilattices.

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] Abuhlail , J. , Morita contexts for corings and equivalence, in "Hopf algebras in noncommutative geometry and physics" (S. Caenepeel and F. Van Oystaeyen, eds.), Lect. Notes Pure Appl. Math. 239 , Dekker, New York , 2004

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] H aghany , A. , Hopfcity and co-Hopfcity for Morita contexts , Comm. Algebra , 27 ( 1999 ), no. 1 , 477 – 492 . [20] H olland , W. C. , M artinez , J

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