Operation is Left Operation iff Anticommutative with Right Cancellable Element

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

Then:
 * $\circ$ is the left operation


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

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:

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

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

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

As $\struct {S, \circ}$ is a semigroup it follows from 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:


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