Abstract
We ascertain conditions and structures on categories and semigroups which admit the construction of pseudo-products and trace products respectively, making their connection as precise as possible. This topic is modelled on the ESN Theorem and its generalisation to ample semigroups. Unlike some other variants of ESN, it is self-dual (two-sided), and the condition of commuting projections is relaxed. The condition that projections form a band (are closed under multiplication) is shown to be a very natural one. One-sided reducts are considered, and compared to (generalised) D-semigroups. Finally the special case when the category is a groupoid is examined.
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The authors thank a referee and the editor for the improvements resulting from (respectively) their gently pointing out a nonsensical passage and suggesting additional references. For sharing some of their work before publication, they thank Mark Lawson and Tim Stokes, who also made helpful comments on drafts of this work including detecting errors.
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Communicated by Victoria Gould.
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FitzGerald, D.G., Kinyon, M.K. Trace- and pseudo-products: restriction-like semigroups with a band of projections. Semigroup Forum 103, 848–866 (2021). https://doi.org/10.1007/s00233-021-10221-x
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DOI: https://doi.org/10.1007/s00233-021-10221-x