Set of Associating Elements forms Subsemigroup of Magma

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:

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.

Let $x, y \in T$.

Let $a, b \in T$.

Then:

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

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.