Characteristic of Cayley Table of Left Operation
Jump to navigation
Jump to search
Theorem
Let $S$ be a finite set.
Let $\leftarrow$ denote the left operation on $S$.
The Cayley table of the algebraic structure $\struct {S, \leftarrow}$ is characterised by the fact that each row contains just one distinct element.
Proof
A row of a Cayley table headed by $x$ contains all those elements of the form $x \leftarrow y$.
By definition of the left operation:
- $x \leftarrow y = x$
Hence the result.
$\blacksquare$
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.9$