\begin{definition}[Definition:Coset/Left Coset]
Let $\struct {S, \circ}$ be an algebraic structure.
Let $\struct {H, \circ}$ be a subgroup of $\struct {S, \circ}$.
The '''left coset of $x$ modulo $H$''', or '''left coset of $H$ by $x$''', is:
:$x \circ H = \set {y \in S: \exists h \in H: y = x \circ h}$
That is, it is the subset product with singleton:
:$x \circ H = \set x \circ H$
\end{definition}