Cosets of Monoids: Conditions for Partitions

[mnb96]

Hi,

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?

 

[Stephen Tashi]

Let M_1 be a monoid and suppose there exists a monoid homomorphism f: M_1 \rightarrow M_2 from M_1 onto another monoid M_2. Let H be the kernel of f 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 M_1. I think those classes are analagous to cosets.