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:

\(\displaystyle x \circ y \circ z \circ x \circ z\) \(=\) \(\displaystyle x \circ \left({y \circ z}\right) \circ x \circ z\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle x \circ z\) $\quad$ $\quad$


Also:

\(\displaystyle x \circ y \circ z \circ x \circ z\) \(=\) \(\displaystyle x \circ y \circ \left({z \circ x \circ z}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle x \circ y \circ z\) $\quad$ $\quad$

Hence the result.

$\blacksquare$


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