Set of Associating Elements forms Subsemigroup of Magma

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Theorem

Let $S$ be a set.

Let $\oplus$ be an operation on $S$ such that $\struct {S, \oplus}$ is a magma.


Let $T \subseteq S$ be the subset of $S$ defined as:

$T = \set {x \in S: \paren {x \oplus y} \oplus z = x \oplus \paren {y \oplus z} }$


Suppose $T \ne \O$.

Then $\struct {T, \oplus {\restriction_T} }$ is a subsemigroup of $\struct {S, \oplus}$.


Proof

Taking the semigroup axioms in turn:


Semigroup Axiom $\text S 1$: Associativity

Because $T$ consists only of elements $x$ such that $\paren {x \oplus y} \oplus z = x \oplus \paren {y \oplus z}$, it follows that $\oplus$ is associative on $T$.

That is, $\struct {T, \oplus {\restriction_T} }$ is associative.

$\Box$


Semigroup Axiom $\text S 0$: Closure

Let $x, y \in T$.

Let $a, b \in T$.

Then:

\(\ds \forall a, b \in T: \, \) \(\ds \paren {\paren {x \oplus y} \oplus a} \oplus b\) \(=\) \(\ds \paren {x \oplus \paren {y \oplus a} } \oplus b\) as $x \in T$
\(\ds \) \(=\) \(\ds x \oplus \paren {\paren {y \oplus a} \oplus b}\) as $x \in T$
\(\ds \) \(=\) \(\ds x \oplus \paren {y \oplus \paren {a \oplus b} }\) as $y \in T$
\(\ds \) \(=\) \(\ds \paren {x \oplus y} \oplus \paren {a \oplus b}\) as $x \in T$

Thus $x \oplus y \in T$ and so $\struct {T, \oplus {\restriction_T} }$ is closed.

$\Box$


The semigroup axioms are thus seen to be fulfilled, and so $\struct {T, \oplus {\restriction_T} }$ is a semigroup.

The result follows by definition of subsemigroup.

$\blacksquare$


Sources

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