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Directory:Jon Awbrey/Papers/Differential Logic (view source)
Revision as of 14:32, 10 June 2009
, 14:32, 10 June 2009→Note 18: numbered list → table + more spacing for ease of reading
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math> It is from this functional perspective that we can see an easy way to derive the two regular representations. Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math> It is from this functional perspective that we can see an easy way to derive the two regular representations. Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
{| align="center" cellpadding="6" width="90%"
| valign="top" | 1.
| <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>x \cdot y.</math>
|-
| valign="top" | 2.
| <math>\operatorname{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>y \cdot x.</math>
|}
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>x \cdot y,</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : x \cdot y) ~|~ y \in G \}.</math> The pairs <math>(y : x \cdot y)</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run across the top margin. This aspect of pragmatic definition we recognize as the regular ante-representation:
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>x \cdot y,</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : x \cdot y) ~|~ y \in G \}.</math> The pairs <math>(y : x \cdot y)</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run across the top margin. This aspect of pragmatic definition we recognize as the regular ante-representation:→
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