# Cosets of Monoids: Conditions for Partitions

• mnb96
In summary, the cosets of a submonoid in a monoid do not always form a partition of the set, but when the submonoid is the kernel of a monoid homomorphism, the equivalence classes of the corresponding equivalence relation still form a partition of the set.
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:
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.

## 1. What is a coset of a monoid?

A coset of a monoid is a set of elements that are obtained by multiplying a fixed element of the monoid by all elements of the monoid. These elements are not necessarily distinct, and the coset forms a subgroup of the monoid.

## 2. What is the significance of cosets of monoids?

Cosets of monoids play an important role in understanding the structure of a monoid. They allow us to partition the monoid into smaller, more manageable subsets and can help us identify important properties of the monoid.

## 3. What are the conditions for a partition formed by cosets of a monoid?

The conditions for a partition formed by cosets of a monoid are that the cosets must be disjoint (meaning that they do not share any elements) and that their union must equal the entire monoid.

## 4. How are cosets of monoids related to normal subgroups?

A coset of a monoid is a special case of a normal subgroup, where the monoid is commutative. In general, a subgroup is considered normal if all left and right cosets of the subgroup are also subgroups of the original group.

## 5. Can cosets of monoids be used in other areas of mathematics?

Yes, cosets of monoids have applications in various areas of mathematics, including group theory, algebraic geometry, and abstract algebra. They are also used in computer science, specifically in the study of finite state machines and automata theory.

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