Changes

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:

# <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>

# <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>

| <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>

| <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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