Characteristic of Cayley Table of Right Operation

Theorem
Let $S$ be a finite set.

Let $\rightarrow$ denote the right operation on $S$.

The Cayley table of the algebraic structure $\struct {S, \rightarrow}$ is characterised by the fact that each column contains just one distinct element.

Proof
A column of a Cayley table headed by $y$ contains all those elements of the form $x \rightarrow y$.

By definition of the right operation:
 * $x \rightarrow y = y$

Hence the result.

Also see

 * Characteristic of Cayley Table of Left Operation