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Morita invariants for semigroups with local units. (English) Zbl 1256.20054

Semigroups \(S\) and \(T\) are strongly Morita equivalent [S. Talwar, J. Algebra 181, No. 2, 371-394 (1996; Zbl 0855.20054)] if there are unitary bi-acts \(_SP_T\) and \(_TQ_S\) and surjective bi-act morphisms \(_S(P\otimes_TQ)_S\to{_SS_S}\) and \(_T(Q\otimes_SP)_T\to{_TT_T}\) that satisfy natural commutativity conditions.
The authors show first that periodicity is always preserved under this equivalence. M. V. Lawson [J. Pure Appl. Algebra 215, No. 4, 455-470 (2011; Zbl 1229.20060)] showed that regularity is preserved under the hypothesis of ‘local units’: that for both semigroups, every element possesses a left and a right unit. This condition is weakened here to ‘weak local units’ (\(s\in sS\cap Ss\) for all \(s\in S\), and likewise for \(T\)) and this and slightly weaker conditions are investigated with regard to a variety of other properties. For instance, if the two semigroups have ‘common two-sided weak units’, then they satisfy the same identities.

MSC:

20M10 General structure theory for semigroups
20M50 Connections of semigroups with homological algebra and category theory
20M30 Representation of semigroups; actions of semigroups on sets
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