Changes

The larger purpose of this discussion is to serve as an introduction, not just to the special topic of devising IRs for sign relations, but to the general issue of producing, using, and comprehending IRs for any kind of relation or any domain of formal objects.  It is hoped that a careful study of these simple IRs can inaugurate a degree of insight into the broader arenas of formalism of which they occupy an initial niche and into the wider landscapes of discourse of which they inhabit a natural corner, in time progressing up to the axiomatic presentation of formal theories about combinatorial domains and other mathematical objects.

The larger purpose of this discussion is to serve as an introduction, not just to the special topic of devising IRs for sign relations, but to the general issue of producing, using, and comprehending IRs for any kind of relation or any domain of formal objects.  It is hoped that a careful study of these simple IRs can inaugurate a degree of insight into the broader arenas of formalism of which they occupy an initial niche and into the wider landscapes of discourse of which they inhabit a natural corner, in time progressing up to the axiomatic presentation of formal theories about combinatorial domains and other mathematical objects.

For the sake of maximum clarity and re-usability of results, I begin by articulating the abstract skeleton of the paradigm structure, treating the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as sundry aspects of a single, unitary, but still uninterpreted object.  Then I return at various successive stages to differentiate and individualize the two interpreters, to arrange more functional flesh on the basis provided by their structural bones, and to illustrate how their bare forms can be arrayed in many different styles of qualitative detail.

For the sake of maximum clarity and reusability of results, I begin by articulating the abstract skeleton of the paradigm structure, treating the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as sundry aspects of a single, unitary, but still uninterpreted object.  Then I return at various successive stages to differentiate and individualize the two interpreters, to arrange more functional flesh on the basis provided by their structural bones, and to illustrate how their bare forms can be arrayed in many different styles of qualitative detail.

In building connections between ERs and IRs of sign relations the discussion turns on two types of partially ordered sets, or ''posets''.  Suppose that <math>L\!</math> is one of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> and let <math>\operatorname{ER}(L)\!</math> be an ER of <math>L.\!</math>

In building connections between ERs and IRs of sign relations the discussion turns on two types of partially ordered sets, or ''posets''.  Suppose that <math>L\!</math> is one of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> and let <math>\operatorname{ER}(L)\!</math> be an ER of <math>L.\!</math>

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W  =  { A,  B, "A", "B", "i", "u"}.

W  =  { A,  B, "A", "B", "i", "u"}.

W  =  {w1, w2,  w3,  w4,  w5,  w6).

W  =  {w1, w2,  w3,  w4,  w5,  w6).