# Cosets of monoids

1. May 26, 2015

### 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?

Last edited: May 26, 2015
2. May 27, 2015

### 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.