Power Structure of Semigroup is Semigroup
Jump to navigation
Jump to search
Theorem
Let $\struct {S, \circ}$ be a semigroup.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Then $\struct {\powerset S, \circ_\PP}$ is a semigroup.
Proof
From Power Structure of Magma is Magma we conclude that $\struct {\powerset S, \circ_\PP}$ is a magma.
It follows from Subset Product within Semigroup is Associative that $\circ_\PP$ is associative in $\struct {\powerset S, \circ_\PP}$.
Thus $\struct {\powerset S, \circ_\PP}$ is a semigroup.
$\blacksquare$