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<math>\begin{array}{c}

<math>\begin{matrix}

f_{1}

f_{1}

\\[4pt]

\\[4pt]

\\[4pt]

\\[4pt]

f_8

f_8

\end{array}\!</math>

\end{matrix}\!</math>

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<math>\begin{matrix}

<math>\begin{matrix}

Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c\!</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}~\!</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a\!</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.

Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c\!</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}~\!</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a\!</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.

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:

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:&nbsp; <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>&nbsp; This is the mode of reading that we call &ldquo;multiplying on the left&rdquo;.

==Logical Cacti==

==Logical Cacti==