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.

$\blacksquare$

Also see

• Characteristic of Cayley Table of Left Operation

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.9$