Changes

===6.6. Basic Notions of Group Theory===

===6.6. Basic Notions of Group Theory===

Many of the most salient themes that have a call to be played out in this work — the application of generic forms of operation to themselves and to each other, the relationship of invariant forms to their variant presentations, and the relationship of abstract forms to their concrete representations — all of these topics arise in a very instructive way within the mathematical subject of group theory.  This is most likely due to the fact that group theory, as a mathematical tool, got its start and much of its later sharpening in the process of trying to clarify the physical and formal phenomena that involve these very same issues.

Many of the most salient themes that have a call to be played out in this work — the application of generic forms of operation to themselves and to each other, the relationship of invariant forms to their variant presentations, and the relationship of abstract forms to their concrete representations — all of these topics arise in a very instructive way within the mathematical subject of group theory.  This is most likely due to the fact that group theory, as a mathematical tool, got its start and much of its later sharpening in the process of trying to clarify the physical and formal phenomena that involve these very same issues.

In group theory, fortunately, these themes arise in a slightly plainer fashion, and the otherwise mystifying questions they involve have been studied to the point that their original mysteries are barely observed.  Thus, a good way to approach the construction of a RIF is to study the well understood versions of self application and self explanation that turn up in group theory.  Given the simpler character and the familiar condition of these topics in that area, they supply a convenient basis for subsequent extensions and help to arrange a staging ground for the types of sign theoretic generalizations that are ultimately desired.

In group theory, fortunately, these themes arise in a slightly plainer fashion, and the otherwise mystifying questions they involve have been studied to the point that their original mysteries are barely observed.  Thus, a good way to approach the construction of a RIF is to study the well understood versions of self-application and self-explanation that turn up in group theory.  Given the simpler character and the familiar condition of these topics in that area, they supply a convenient basis for subsequent extensions and help to arrange a staging ground for the types of sign theoretic generalizations that are ultimately desired.

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 "series 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 "<Xi>" is assumed by default to refer to a SOD <Xi>, where each Xi is assumed to be a nonempty set.

A "series 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 "<Xi>" is assumed by default to refer to a SOD <Xi>, where each Xi is assumed to be a nonempty set.