Associative Idempotent Anticommutative

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Theorem

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

Let $\circ$ be associative.


Then $\circ$ is anticommutative if and only if:

$(1): \quad \circ$ is idempotent

and:

$(2): \quad \forall a, b \in S: a \circ b \circ a = a$


Proof

Let $\circ$ be an associative operation on $S$.


Necessary Condition

Suppose $\circ$ is anticommutative.

Since $S$ is non-empty, let $a \in S$ be arbitrary.


By definition, $\circ$ is anticommutative implies that:

$\forall a \in S: a \circ a = a \circ a \iff a = a$

By $\circ$ is a well-defined operation, $a \circ a$ is well-defined.


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It remains to be shown that $a \circ a \in S$.


Let $a, w \in S$ be arbitrary.

Since $\circ$ is anticommutative, then:

$\paren {a \circ a} \circ w = w \circ \paren {a \circ a} \iff a \circ a = w$

Thus $a \circ a \in S$.

That is:

$\forall \paren {a \circ a}, w \in S: \paren {a \circ a} \circ w = w \circ \paren {a \circ a} \iff a \circ a = w$


Let $w = a \in S$.

Then:

$\forall \paren {a \circ a}, a \in S: \paren {a \circ a} \circ a = a \circ \paren {a \circ a} \iff a \circ a = a$

Hence:

$\forall a \in S: \paren {a \circ a} \circ a = a \circ \paren {a \circ a} \iff a \circ a = a$

So $\circ$ being associative and anticommutative implies that $\circ$ is idempotent.


Now, it remains to be shown that for any $a, b \in S$, if $\circ$ is associative and anticommutative, then:

$ a \circ b \circ a = a$.

In particular, it remains to be shown that:

$a \circ b \circ a \in S$.

By assumption:

$\forall a, b \in S: a \circ b = b \circ a \iff a = b$

By $\circ$ is a well-defined operation, $a \circ b$ and $b \circ a$ are well-defined.


Let $a, b, x \in S$, then:

$\paren {a \circ b} \circ x = x \circ \paren {a \circ b} \iff a \circ b = x$


This implies that:

$\forall \paren {a \circ b}, x \in S: \paren {a \circ b} \circ x = x \circ \paren {a \circ b} \iff a \circ b = x$


Hence there exists $\paren {a \circ b} \in S$ for some $a, b \in S$.


Similarly, let $b, a, y \in S$, then:

$\paren {b \circ a} \circ y = y \circ \paren {b \circ a} \iff b \circ a = y$


This implies that:

$\forall \paren {b \circ a}, y \in S: \paren {b \circ a} \circ y = y \circ \paren {b \circ a} \iff b \circ a = y$


Hence there exists $\paren {b \circ a} \in S$ for some $a, b \in S$.


Let $a, b, z \in S$, then:

$\paren {a \circ b \circ a} \circ z = z \circ \paren {a \circ b \circ a} \iff a \circ b \circ a = z$


Hence there exists $\paren {a \circ b \circ a} \in S$ for some $a, b \in S$.

Hence:

\(\ds \paren {a \circ a} \circ b \circ a\) \(=\) \(\ds a \circ b \circ a\) $\circ$ is idempotent, proved above
\(\ds \) \(=\) \(\ds a \circ b \circ \paren {a \circ a}\) $\circ$ is idempotent
\(\ds \leadsto \ \ \) \(\ds a \circ \paren {a \circ b \circ a}\) \(=\) \(\ds \paren {a \circ b \circ a} \circ a\) $\circ$ is associative
\(\ds \leadsto \ \ \) \(\ds a \circ b \circ a\) \(=\) \(\ds a\) $\circ$ is anticommutative
\(\ds \leadsto \ \ \) \(\ds a \circ b \circ a\) \(=\) \(\ds a\) by hypothesis


So $\circ$ being associative and anticommutative implies, via the fact (also proved) that $\circ$ is idempotent, that $a \circ b \circ a = a$.

$\Box$


Sufficient Condition

Now, suppose $\circ$ (which we take to be associative) is idempotent and:

$\forall a, b \in S: a \circ b \circ a = a$

It remains to be shown that $\circ$ is anticommutative.


Suppose:

$a \circ b = b \circ a$.

Then:

\(\ds \leadsto \ \ \) \(\ds a \circ \paren {b \circ a}\) \(=\) \(\ds a \circ \paren {a \circ b}\) by hypothesis: $a \circ b \circ a = a\in S$
\(\ds a \circ b \circ a\) \(=\) \(\ds \paren {a \circ a} \circ b\) $\circ$ is associative
\(\ds \) \(=\) \(\ds a \circ b\) $\circ$ is idempotent
\(\ds \) \(=\) \(\ds a\) by hypothesis


Similarly:

\(\ds a \circ b\) \(=\) \(\ds b \circ a\)
\(\ds \leadsto \ \ \) \(\ds \paren {a \circ b} \circ a\) \(=\) \(\ds \paren {b \circ a} \circ a\) by hypothesis: $a \circ b \circ a \in S$
\(\ds a \circ b \circ a\) \(=\) \(\ds b \circ \paren {a \circ a}\) $\circ$ is associative
\(\ds \) \(=\) \(\ds b \circ a\) $\circ$ is idempotent
\(\ds \) \(=\) \(\ds a\) by hypothesis


Also:

\(\ds a \circ b\) \(=\) \(\ds b \circ a\)
\(\ds \leadsto \ \ \) \(\ds b \circ \paren {a \circ b}\) \(=\) \(\ds b \circ \paren {b \circ a}\) by hypothesis: $b \circ a \circ b = b \in S$
\(\ds b \circ a \circ b\) \(=\) \(\ds \paren {b \circ b} \circ a\) $\circ$ is associative
\(\ds \) \(=\) \(\ds b \circ a\) $\circ$ is idempotent
\(\ds \) \(=\) \(\ds b\) by hypothesis


Then:

\(\ds b \circ a\) \(=\) \(\ds a \circ b\)
\(\ds \leadsto \ \ \) \(\ds \paren {b \circ a} \circ b\) \(=\) \(\ds \paren {a \circ b} \circ b\) by hypothesis: $b \circ a \circ b = b \in S$
\(\ds \leadsto \ \ \) \(\ds b \circ a \circ b\) \(=\) \(\ds a \circ \paren {b \circ b}\) $\circ$ is associative
\(\ds \) \(=\) \(\ds a \circ b\) $\circ$ is idempotent
\(\ds \) \(=\) \(\ds b\) by hypothesis


Hence:

\(\ds a\) \(=\) \(\ds a \circ b = b\)
\(\ds a\) \(=\) \(\ds b \circ a = b\)
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds b\)


So:

$(1): \quad \circ$ is idempotent

and:

$(2): \quad \forall a, b \in S: a \circ b \circ a = a$

together imply that $\circ$ is anticommutative.

$\blacksquare$


Sources

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