Operation is Right Operation iff Anticommutative with Left Cancellable Element

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

Then:
 * $\circ$ is the right operation


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

Sufficient Condition
Let $\circ$ be the right operation.

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

Let $x \in S$ be arbitrary.

Let $y, z \in S$ such that:
 * $x \circ z = x \circ y$

Then:

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

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

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

As $\struct {S, \circ}$ it follows a priori that $\circ$ is associative.

Hence from Associative and Anticommutative:


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

As $z$ is left cancellable:


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