Changes

==Note 14==

==Note 14==

The next few excursions in this series will provide a scenic tour of various ideas in group theory that will turn out to be of constant guidance in several of the settings that are associated with our topic.

The next few excursions in this series will provide

a scenic tour of various ideas in group theory that

will turn out to be of constant guidance in several

of the settings that are associated with our topic.

Let me return to Peirce's early papers on the algebra of relatives

Let me return to Peirce's early papers on the algebra of relatives to pick up the conventions that he used there, and then rewrite my account of regular representations in a way that conforms to those.

to pick up the conventions that he used there, and then rewrite my

account of regular representations in a way that conforms to those.

Peirce expresses the action of an "elementary dual relative" like so:

Peirce describes the action of an "elementary dual relative" in this way:

| [Let] A:B be taken to denote

{| align="center" cellpadding="6" width="90%"

| the elementary relative which

| Elementary simple relatives are connected together in systems of four.  For if <math>\mathrm{A}\!:\!\mathrm{B}</math> be taken to denote the elementary relative which multiplied into <math>\mathrm{B}\!</math> gives <math>\mathrm{A},\!</math> then this relation existing as elementary, we have the four elementary relatives

| multiplied into B gives A.

|

| align="center" | <math>\mathrm{A}\!:\!\mathrm{A} \qquad \mathrm{A}\!:\!\mathrm{B} \qquad \mathrm{B}\!:\!\mathrm{A} \qquad \mathrm{B}\!:\!\mathrm{B}.</math>

| Peirce, 'Collected Papers', CP 3.123.

|-

| C.S. Peirce, ''Collected Papers'', CP&nbsp;3.123.

Peirce is well aware that it is not at all necessary to arrange the

Peirce is well aware that it is not at all necessary to arrange the

elementary relatives of a relation into arrays, matrices, or tables,

elementary relatives of a relation into arrays, matrices, or tables,