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Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call "multiplying on the left".
Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call "multiplying on the left".
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<pre>
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In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling <math>i\!</math> the relate and <math>j\!</math> the correlate, the elementary relative <math>i\!:\!j</math> now means that <math>i\!</math> gets changed into <math>j.\!</math> In this scheme of reading, the transformation <math>a\!:\!b + b\!:\!c + c\!:\!a</math> is a permutation of the aggregate <math>\mathbf{1} = a + b + c,</math> or what we would now call the set <math>\{ a, b, c \},\!</math> in particular, it is the permutation that is otherwise notated as follows:
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In the algebraic, permutational, or transformational contexts of
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application, however, Peirce converts to the alternative mode of
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reading, although still calling i the relate and j the correlate,
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the elementary relative i:j now means that i gets changed into j.
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In this scheme of reading, the transformation a:b + b:c + c:a is
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a permutation of the aggregate $1$ = a + b + c, or what we would
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now call the set {a, b, c}, in particular, it is the permutation
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that is otherwise notated as:
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( a b c )
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{| align="center" cellpadding="6"
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< >
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|
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( b c a )
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<math>\begin{Bmatrix}
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a & b & c
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\\
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b & c & a
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\end{Bmatrix}</math>
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|}
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This is consistent with the convention that Peirce uses in
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This is consistent with the convention that Peirce uses in the paper "On a Class of Multiple Algebras" (CP 3.324–327).
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the paper "On a Class of Multiple Algebras" (CP 3.324-327).
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</pre>
==Note 16==
==Note 16==