Abstract
Two semigroups are called Morita equivalent if the categories of firm right acts over them are equivalent. We prove that every semigroup is Morita equivalent to its subsemigroup consisting of all products of n factors. Using this we show that a finite semigroup is Morita equivalent to its largest factorisable subsemigroup. It follows that two finite semigroups are Morita equivalent if and only if their Cauchy completions are equivalent categories. Since these categories are finite, the problem of Morita equivalence of finite semigroups is decidable.
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Communicated by Mark V. Lawson.
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Research of Ü. Reimaa and V. Laan was partially supported by the Estonian Research Council grant PUT1519, research of L. Tart was supported by the Estonian Institutional Research Project IUT20-57. The work of the first author was supported in part by the Estonian Research Council Grants PRG49 and PSG114.
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Reimaa, Ü., Laan, V. & Tart, L. Morita equivalence of finite semigroups. Semigroup Forum 102, 842–860 (2021). https://doi.org/10.1007/s00233-020-10153-y
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DOI: https://doi.org/10.1007/s00233-020-10153-y