Coset

Definition 1. Let $M$ be a magma, $A$ be a submagma and $m \in M$.

• The left coset of $A$ by $m$ is

  \[mA = \{ma : a \in A\}.\]

  The left coset quotient of $M$ by $A$ is

  \[M / A = \{mA : m \in M\}.\]

• The right coset of $A$ by $m$ is

  \[Am = \{am : a \in A\}.\]

  The right coset quotient of $M$ by $A$ is

  \[M \backslash A = \{Am : m \in M\}.\]

Proposition 1. Let $M$ be an abelian magma and $A$ be a submagma. For any $m \in M$,

\[mA = Am,\]

and hence

\[M / A = M \backslash A.\]