Structure under Left Operation is Semigroup

Theorem
Let $\struct {S, \gets}$ be an algebraic structure in which the operation $\gets$ is the left operation.

Then $\struct {S, \gets}$ is a semigroup.

Proof
We need to verify the semigroup axioms:

By the nature of the right operation, $\struct {S, \to}$ is closed:
 * $\forall x, y \in S: x \gets y = y \in S$

whatever $S$ may be.

Hence holds.

From Right Operation is Associative, $\gets$ is associative.

Hence holds.

So $\struct {S, \gets}$ is a semigroup.

Also see

 * Structure under Right Operation is Semigroup