Set of Subsemigroups of Commutative Semigroup form Subsemigroup of Power Structure
Theorem
Let $\struct {S, \circ}$ be a commutative semigroup.
Let $\struct {\powerset S, \circ_\PP}$ denote the power structure of $\struct {S, \circ}$.
Let $\TT$ be the set of all subsemigroups of $S$.
Then $\struct {\TT, \circ_\PP}$ is a subsemigroup of $\struct {\powerset S, \circ_\PP}$.
Proof
First we establish that from Power Structure of Semigroup is Semigroup:
- $\struct {\powerset S, \circ_\PP}$ is a semigroup.
From Subset Product within Commutative Structure is Commutative:
- $\struct {\powerset S, \circ_\PP}$ is a commutative semigroup.
Let $A$ and $B$ be arbitrary subsemigroups of $S$.
As $A$ and $B$ are subsemigroups of $S$, they themselves are closed for $\circ$.
That is:
- $\forall x, y \in A: x \circ y \in A$
and:
- $\forall x, y \in B: x \circ y \in B$
By definition of operation induced on $\powerset S$:
- $A \circ_\PP B = \set {a \circ b: a \in A, b \in B}$
We are to show that:
- $\forall x, y \in A \circ_\PP B: x \circ y \in A \circ_\PP B$
Let $x, y \in A \circ_\PP B$ such that $x = a_x \circ b_x$, $y = a_y \circ b_y$.
We have:
\(\ds x \circ y\) | \(=\) | \(\ds \paren {a_x \circ b_x} \circ \paren {a_y \circ b_y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a_x \circ \paren {b_x \circ a_y} \circ b_y\) | Semigroup Axiom $\text S 1$: Associativity | |||||||||||
\(\ds \) | \(=\) | \(\ds a_x \circ \paren {a_y \circ b_x} \circ b_y\) | $\circ$ is commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a_x \circ a_y} \circ \paren {b_x \circ b_y}\) | Semigroup Axiom $\text S 1$: Associativity | |||||||||||
\(\ds \) | \(\in\) | \(\ds A \circ_\PP B\) | Semigroup Axiom $\text S 0$: Closure: both $A$ and $B$ are closed for $\circ$ |
That is:
- $x, y \in A \circ_\PP B \implies x \circ y \in A \circ_\PP B$
and $A \circ_\PP B$ is seen to be $\struct {\TT, \circ_\PP}$.
Hence the result.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets: Exercise $9.9$