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This section develops the aspects of group theory that are needed in this work, bringing together a fundamental selection of abstract ideas and concrete examples that are used repeatedly throughout the rest of the project.  To start, I present an abstract formulation of the basic concepts of group theory, beginning from a very general setting in the theory of relations and proceeding in quick order to the definitions of groups and their representations.  After that, I describe a couple of concrete examples that are designed mainly to illustrate the abstract features of groups, but that also appear in different guises at later stages of this discussion.

This section develops the aspects of group theory that are needed in this work, bringing together a fundamental selection of abstract ideas and concrete examples that are used repeatedly throughout the rest of the project.  To start, I present an abstract formulation of the basic concepts of group theory, beginning from a very general setting in the theory of relations and proceeding in quick order to the definitions of groups and their representations.  After that, I describe a couple of concrete examples that are designed mainly to illustrate the abstract features of groups, but that also appear in different guises at later stages of this discussion.

A '''sequence of domains''' (SOD) is a nonempty sequence of nonempty sets.  A declarative indication of a sequence of sets, typically offered in staking out the grounds of a discussion, is taken for granted as a SOD.  Thus, the notation <math>{}^{\backprime\backprime}(X_i)_i{}^{\prime\prime}</math> is assumed by default to refer to a SOD <math>(X_i)_i,\!</math> where each <math>X_i\!</math> is assumed to be a nonempty set.

A '''sequence of domains''' (SOD) is a nonempty sequence of nonempty sets.  A declarative indication of a sequence of sets, typically offered in staking out the grounds of a discussion, is taken for granted as a SOD.  Thus, the notation <math>{}^{\backprime\backprime}(X_i){}^{\prime\prime}</math> is assumed by default to refer to a SOD <math>(X_i)_{i \in I},\!</math> where each <math>X_i\!</math> is assumed to be a nonempty set.

Given a SOD <math>(X_i)_i,\!</math> its cartesian product, notated as <math>\textstyle\prod_i (X_i)</math> or <math>\textstyle\prod_i X_i,</math> is defined as follows:

Given a SOD <math>(X_i),\!</math> its cartesian product, notated as <math>\textstyle\prod_i (X_i)</math> or <math>\textstyle\prod_i X_i,</math> is defined as follows:

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

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

| <math>\prod_i (X_i) = \prod_i X_i = \{ (x_i)_i : x_i \in X_i \}.</math>

| <math>\prod_i (X_i) = \prod_i X_i = \{ (x_i) : x_i \in X_i \}.</math>

|}

|}

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An "n ary relation" or an "n place relation" is a relation on an ordered n tuple of nonempty sets.  Thus, R is an n place relation on the SOD <X1, ..., Xn> if and only if R c X1x...xXn.  In various applications, the n tuple elements <x1, ..., xn> of R are called its "elementary relations", "individual transactions", "ingredients", or "effects".

An "n ary relation" or an "n place relation" is a relation on an ordered n tuple of nonempty sets.  Thus, R is an n place relation on the SOD <X1, ..., Xn> if and only if R c X1x...xXn.  In various applications, the n tuple elements <x1, ..., xn> of R are called its "elementary relations", "individual transactions", "ingredients", or "effects".