Operation is Left Operation iff Anticommutative with Right Cancellable Element

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Theorem

Let $\struct {S, \circ}$ be an semigroup.

Then:

$\circ$ is the left operation

if and only if:

$\circ$ is anticommutative and has a right cancellable element.


Proof

Sufficient Condition

Let $\circ$ be the left operation.

Then from Left Operation is Anticommutative we have that $\circ$ is anticommutative.


Let $x \in S$ be arbitrary.

Let $y, z \in S$ such that:

$z \circ x = y \circ x$


Then:

\(\ds z \circ x\) \(=\) \(\ds z\) Definition of Left Operation
\(\ds y \circ x\) \(=\) \(\ds y\) Definition of Left Operation
\(\ds \leadsto \ \ \) \(\ds z\) \(=\) \(\ds y\)

That is, $x$ is a right cancellable element for all $x \in S$.


Thus:

$\circ$ is anticommutative and has a right cancellable element.

$\Box$


Necessary Condition

Let $\circ$ be anticommutative and have a right cancellable element $z$.

As $\struct {S, \circ}$ is a semigroup it follows from Semigroup Axiom $\text S 1$: Associativity that $\circ$ is associative.


Hence from Associative and Anticommutative:

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

As $z$ is right cancellable:

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

$\blacksquare$


Sources