Embeddings in coset monoids

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    Abstract

    A submonoid S of a monoid M is said to be cofull if it contains the group of units of M. We extract from the work of Easdown, East and FitzGerald (2002) a sufficient condition for a monoid to embed as a cofull submonoid of the coset monoid of its group of units, and show further that this condition is necessary. This yields a simple description of the class of finite monoids which embed in the coset monoids of their group of units. We apply our results to give a simple proof of the result of McAlister [D. B. McAlister, ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum 20 (1980), 255–267] which states that the symmetric inverse semigroup on a finite set X does not embed in the coset monoid of the symmetric group on X. We also explore examples, which are necessarily infinite, of embeddings whose images are not cofull.
    Original languageEnglish
    Pages (from-to)75-80
    Number of pages6
    JournalJournal of the Australian Mathematical Society
    Volume85
    Issue number1
    DOIs
    Publication statusPublished - 2008

    Keywords

    • monoids
    • semigroups

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