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Cosets of monoids

  1. May 26, 2015 #1

    We know that given a group G and a subgroup H, the cosets of H in G partition the set G.
    Now, if instead of groups we consider a monoid M and a submonoid H, the cosets of H in M in general do not partition the set M.

    However, are there some conditions that we can impose on H under which its cosets still form a partition of M?
    Last edited: May 26, 2015
  2. jcsd
  3. May 27, 2015 #2

    Stephen Tashi

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    Let [itex] M_1 [/itex] be a monoid and suppose there exists a monoid homomorphism [itex] f: M_1 \rightarrow M_2 [/itex] from [itex] M_1 [/itex] onto another monoid [itex] M_2 [/itex]. Let [itex]H [/itex] be the kernel of [itex]f [/itex] as a set. There is also a definition of "kernel" that defines it as an equivalence relation. (http://en.wikipedia.org/wiki/Kernel_(set_theory)) The equivalence classes of that equivalence relation partition partition [itex] M_1 [/itex]. I think those classes are analagous to cosets.
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