Associative and Anticommutative

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Theorem

Let $\circ$ be a binary operation on a set $S$.

Let $\circ$ be both associative and anticommutative.


Then:

$\forall x, y, z \in S: x \circ y \circ z = x \circ z$


Proof

Let $\circ$ be both associative and anticommutative.

Then from Associative Idempotent Anticommutative:

$\forall x, z \in S: x \circ z \circ x = x$

and $\circ$ is idempotent.


Consider $x \circ y \circ z \circ x \circ z$.

We have:

\(\ds x \circ y \circ z \circ x \circ z\) \(=\) \(\ds x \circ \paren {y \circ z} \circ x \circ z\)
\(\ds \) \(=\) \(\ds x \circ z\)


Also:

\(\ds x \circ y \circ z \circ x \circ z\) \(=\) \(\ds x \circ y \circ \paren {z \circ x \circ z}\)
\(\ds \) \(=\) \(\ds x \circ y \circ z\)

Hence the result.

$\blacksquare$


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