Definition:Subband

Definition
Let $\struct {S, \circ}$ be an algebraic structure.

Let $T \subseteq S$ such that $\struct {T, \circ {\restriction_T} }$ is a band, where $\circ {\restriction_T}$ denotes the restriction of $\circ$ to $T$.

Then $\struct {T, \circ {\restriction_T} }$ is a subband of $S$.

Notation
It is usual, for the sake of simplicity, for the same symbol to be used for both $\circ$ and its restriction.

Thus we refer to $\struct {T, \circ}$, and we write:


 * $\struct {T, \circ} \subseteq \struct {S, \circ}$