Difference between revisions of "Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6"

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
−
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(A)\!</math>
+
|+ style="height:30px" | <math>\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(A)\!</math>
 
|- style="height:40px; background:#f0f0ff"
 
|- style="height:40px; background:#f0f0ff"
 
| <math>\text{Object}\!</math>
 
| <math>\text{Object}\!</math>
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| valign="bottom" width="33%" |
 
| valign="bottom" width="33%" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & )
 +
\end{matrix}</math>
 +
| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
 
{}^{\langle} \text{A} {}^{\rangle}
 
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\end{matrix}</math>
 +
| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\end{matrix}</math>
 +
|-
 +
| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & )
 +
\end{matrix}</math>
 +
| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
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\end{matrix}</math>
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| valign="bottom" width="33%" |
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<math>\begin{matrix}
 +
{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
 +
\end{matrix}</math>
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|-
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| valign="bottom" width="33%" |
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<math>\begin{matrix}
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( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & )
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\\
 +
( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & )
 +
\end{matrix}</math>
 +
| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
{}^{\langle} \text{B} {}^{\rangle}
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\\
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{}^{\langle} \text{B} {}^{\rangle}
 +
\\
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{}^{\langle} \text{B} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{B} {}^{\rangle}
 +
\end{matrix}</math>
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| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
{}^{\langle} \text{B} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
 
{}^{\langle} \text{B} {}^{\rangle}
 
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle} \text{i} {}^{\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 +
\end{matrix}</math>
 +
|-
 +
| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & )
 
\\
 
\\
−
{}^{\langle} \text{u} {}^{\rangle}
+
( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & )
 
\end{matrix}</math>
 
\end{matrix}</math>
 
| valign="bottom" width="33%" |
 
| valign="bottom" width="33%" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
| valign="bottom" width="33%" |
 
| valign="bottom" width="33%" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\end{matrix}</math>
 
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
−
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(B)\!</math>
+
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(B)\!</math>
 
|- style="height:40px; background:#f0f0ff"
 
|- style="height:40px; background:#f0f0ff"
 
| <math>\text{Object}\!</math>
 
| <math>\text{Object}\!</math>
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| valign="bottom" width="33%" |
 
| valign="bottom" width="33%" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 +
( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & )
 +
\end{matrix}</math>
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| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
 +
\\
 
{}^{\langle} \text{A} {}^{\rangle}
 
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
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{}^{\langle} \text{A} {}^{\rangle}
 +
\end{matrix}</math>
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| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
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{}^{\langle} \text{A} {}^{\rangle}
 +
\\
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{}^{\langle} \text{A} {}^{\rangle}
 +
\end{matrix}</math>
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|-
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| valign="bottom" width="33%" |
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<math>\begin{matrix}
 +
( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & )
 +
\end{matrix}</math>
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| valign="bottom" width="33%" |
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<math>\begin{matrix}
 +
{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
 +
\\
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{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
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\end{matrix}</math>
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| valign="bottom" width="33%" |
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<math>\begin{matrix}
 +
{}^{\langle} \text{A} {}^{\rangle}
 +
\\
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{}^{\langle} \text{A} {}^{\rangle}
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\\
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{}^{\langle} \text{A} {}^{\rangle}
 +
\\
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{}^{\langle} \text{A} {}^{\rangle}
 +
\end{matrix}</math>
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|-
 +
| valign="bottom" width="33%" |
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<math>\begin{matrix}
 +
( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & )
 +
\\
 +
( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & )
 +
\end{matrix}</math>
 +
| valign="bottom" width="33%" |
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<math>\begin{matrix}
 +
{}^{\langle} \text{B} {}^{\rangle}
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\\
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{}^{\langle} \text{B} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{B} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{B} {}^{\rangle}
 +
\end{matrix}</math>
 +
| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
{}^{\langle} \text{B} {}^{\rangle}
 +
\\
 +
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
 
{}^{\langle} \text{B} {}^{\rangle}
 
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle} \text{i} {}^{\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 +
\end{matrix}</math>
 +
|-
 +
| valign="bottom" width="33%" |
 +
<math>\begin{matrix}
 +
( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & )
 +
\\
 +
( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & )
 
\\
 
\\
−
{}^{\langle} \text{u} {}^{\rangle}
+
( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & )
 
\end{matrix}</math>
 
\end{matrix}</math>
 
| valign="bottom" width="33%" |
 
| valign="bottom" width="33%" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
| valign="bottom" width="33%" |
 
| valign="bottom" width="33%" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
+
{}^{\langle} \text{B} {}^{\rangle}
 
\\
 
\\
−
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
+
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|}
 
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Revision as of 15:24, 3 May 2012

Discussion

Scrap Area

Edit Buffer

When it comes to the subject of systems theory, a particular POV is so widely propagated that it might as well be regarded as the established, received, or traditional POV. The POV in question says that there are dynamic systems and symbolic systems, and never the twain shall meet. I naturally intend to challenge this assumption, preferring to suggest that dynamic …

Table Scraps

Table 37.1  Sign Relational Schema C
	Object	Sign	Interpretant
	x	"x"	"x"
	"x"	"x"	"x"

Table 37.2  Sign Relational Schema D
	Object	Sign	Interpretant
	x	"x"	"x"

Table 37.3  Sign Relational Schema E
	Object	Sign	Interpretant
	"x"	"x"	"x"

Table 37.4  Sign Relational Schema D'
	Object	Sign	Interpretant
	x	"x"	"x"
	x	"x"	<x>
	x	<x>	"x"
	x	<x>	<x>

Work Area

Alternate Text

A semigroup consists of a nonempty set with an associative LOC on it. On formal occasions, a semigroup is introduced by means a formula like \(X = (X, *),\!\) interpreted to mean that a semigroup \(X\!\) is specified by giving two pieces of data, a nonempty set that conventionally, if somewhat ambiguously, goes under the same name \({}^{\backprime\backprime} X {}^{\prime\prime},\!\) plus an associative binary operation denoted by \({}^{\backprime\backprime} * {}^{\prime\prime}.\!\) In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the name of the underlying set. In contexts where more than one semigroup is formed on the same set, one may use notations like \(X_i = (X, *_i)\!\) to distinguish them.

Additive Presentation

Version 1

The \(n^\text{th}\!\) multiple of an element \(x\!\) in a semigroup \(\underline{X} = (X, +, 0),\!\) for integer \(n > 0,\!\) is notated as \(nx\!\) and defined as follows. Proceeding recursively, for \(n = 1,\!\) let \(1x = x,\!\) and for \(n > 1,\!\) let \(nx = (n-1)x + x.\!\)
The \(n^\text{th}\!\) multiple of \(x\!\) in a monoid \(\underline{X} = (X, +, 0),\!\) for integer \(n \ge 0,\!\) is defined the same way for \(n > 0,\!\) letting \(0x = 0\!\) when \(n = 0.\!\)
The \(n^\text{th}\!\) multiple of \(x\!\) in a group \(\underline{X} = (X, +, 0),\!\) for any integer \(n,\!\) is defined the same way for \(n \ge 0,\!\) letting \(nx = (-n)(-x)\!\) for \(n < 0.\!\)

Version 2

In a semigroup written additively, the \(n^\text{th}\!\) multiple of an element \(x\!\) is notated as \(nx\!\) and defined for every positive integer \(n\!\) in the following manner. Proceeding recursively, let \(1x = x\!\) and let \(nx = (n-1)x + x\!\) for all \(n > 1.\!\)
In a monoid written additively, the multiple \(nx\!\) is defined for every non-negative integer \(n\!\) by letting \(0x = 0\!\) and proceeding the same way for \(n > 0.\!\)
In a group written additively, the multiple \(nx\!\) is defined for every integer \(n\!\) by letting \(nx = (-n)(-x)\!\) for \(n < 0\!\) and proceeding the same way for \(n \ge 0.\!\)

Table Work

Group Operations


\(\text{Table 32.1}~~\text{Scheme of a Group Operation Table}\)
\(*\!\) \(x_0\!\) \(\cdots\!\) \(x_j\!\) \(\cdots\!\)
\(x_0\!\) \(x_0 * x_0\!\) \(\cdots\!\) \(x_0 * x_j\!\) \(\cdots\!\)
\(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\)
\(x_i\!\) \(x_i * x_0\!\) \(\cdots\!\) \(x_i * x_j\!\) \(\cdots\!\)
\(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\)


\(\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}\)
\(\text{Element}\!\) \(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(x_0\!\) \(\{\!\) \((x_0 ~,~ x_0 * x_0),\!\) \(\cdots\!\) \((x_j ~,~ x_0 * x_j),\!\) \(\cdots\!\) \(\}\!\)
\(\cdots\!\) \(\{\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\}\!\)
\(x_i\!\) \(\{\!\) \((x_0 ~,~ x_i * x_0),\!\) \(\cdots\!\) \((x_j ~,~ x_i * x_j),\!\) \(\cdots\!\) \(\}\!\)
\(\cdots\!\) \(\{\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\}\!\)


\(\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}\)
\(\text{Element}\!\) \(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(x_0\!\) \(\{\!\) \((x_0 ~,~ x_0 * x_0),\!\) \(\cdots\!\) \((x_j ~,~ x_j * x_0),\!\) \(\cdots\!\) \(\}\!\)
\(\cdots\!\) \(\{\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\}\!\)
\(x_i\!\) \(\{\!\) \((x_0 ~,~ x_0 * x_i),\!\) \(\cdots\!\) \((x_j ~,~ x_j * x_i),\!\) \(\cdots\!\) \(\}\!\)
\(\cdots\!\) \(\{\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\cdots\!\) \(\}\!\)


\(\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4\)
\(\cdot\!\) \(\operatorname{e}\) \(\operatorname{f}\) \(\operatorname{g}\) \(\operatorname{h}\)
\(\operatorname{e}\) \(\operatorname{e}\) \(\operatorname{f}\) \(\operatorname{g}\) \(\operatorname{h}\)
\(\operatorname{f}\) \(\operatorname{f}\) \(\operatorname{e}\) \(\operatorname{h}\) \(\operatorname{g}\)
\(\operatorname{g}\) \(\operatorname{g}\) \(\operatorname{h}\) \(\operatorname{e}\) \(\operatorname{f}\)
\(\operatorname{h}\) \(\operatorname{h}\) \(\operatorname{g}\) \(\operatorname{f}\) \(\operatorname{e}\)


\(\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4\)
\(\text{Element}\!\) \(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(\operatorname{e}\) \(\{\!\) \((\operatorname{e}, \operatorname{e}),\) \((\operatorname{f}, \operatorname{f}),\) \((\operatorname{g}, \operatorname{g}),\) \((\operatorname{h}, \operatorname{h})\) \(\}\!\)
\(\operatorname{f}\) \(\{\!\) \((\operatorname{e}, \operatorname{f}),\) \((\operatorname{f}, \operatorname{e}),\) \((\operatorname{g}, \operatorname{h}),\) \((\operatorname{h}, \operatorname{g})\) \(\}\!\)
\(\operatorname{g}\) \(\{\!\) \((\operatorname{e}, \operatorname{g}),\) \((\operatorname{f}, \operatorname{h}),\) \((\operatorname{g}, \operatorname{e}),\) \((\operatorname{h}, \operatorname{f})\) \(\}\!\)
\(\operatorname{h}\) \(\{\!\) \((\operatorname{e}, \operatorname{h}),\) \((\operatorname{f}, \operatorname{g}),\) \((\operatorname{g}, \operatorname{f}),\) \((\operatorname{h}, \operatorname{e})\) \(\}\!\)


\(\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4\)
\(\text{Element}\!\) \(\text{Function as Set of Ordered Pairs of Symbols}\!\)
\(\operatorname{e}\) \(\{\!\) \(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})\) \(\}\!\)
\(\operatorname{f}\) \(\{\!\) \(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})\) \(\}\!\)
\(\operatorname{g}\) \(\{\!\) \(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})\) \(\}\!\)
\(\operatorname{h}\) \(\{\!\) \(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\) \(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})\) \(\}\!\)


\(\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)\)
\(\cdot\!\) \(\operatorname{1}\) \(\operatorname{a}\) \(\operatorname{b}\) \(\operatorname{c}\)
\(\operatorname{1}\) \(\operatorname{1}\) \(\operatorname{a}\) \(\operatorname{b}\) \(\operatorname{c}\)
\(\operatorname{a}\) \(\operatorname{a}\) \(\operatorname{b}\) \(\operatorname{c}\) \(\operatorname{1}\)
\(\operatorname{b}\) \(\operatorname{b}\) \(\operatorname{c}\) \(\operatorname{1}\) \(\operatorname{a}\)
\(\operatorname{c}\) \(\operatorname{c}\) \(\operatorname{1}\) \(\operatorname{a}\) \(\operatorname{b}\)


\(\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)\)
\(\text{Element}\!\) \(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(\operatorname{1}\) \(\{\!\) \((\operatorname{1}, \operatorname{1}),\) \((\operatorname{a}, \operatorname{a}),\) \((\operatorname{b}, \operatorname{b}),\) \((\operatorname{c}, \operatorname{c})\) \(\}\!\)
\(\operatorname{a}\) \(\{\!\) \((\operatorname{1}, \operatorname{a}),\) \((\operatorname{a}, \operatorname{b}),\) \((\operatorname{b}, \operatorname{c}),\) \((\operatorname{c}, \operatorname{1})\) \(\}\!\)
\(\operatorname{b}\) \(\{\!\) \((\operatorname{1}, \operatorname{b}),\) \((\operatorname{a}, \operatorname{c}),\) \((\operatorname{b}, \operatorname{1}),\) \((\operatorname{c}, \operatorname{a})\) \(\}\!\)
\(\operatorname{c}\) \(\{\!\) \((\operatorname{1}, \operatorname{c}),\) \((\operatorname{a}, \operatorname{1}),\) \((\operatorname{b}, \operatorname{a}),\) \((\operatorname{c}, \operatorname{b})\) \(\}\!\)


\(\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)\)
\(+\!\) \(\operatorname{0}\) \(\operatorname{1}\) \(\operatorname{2}\) \(\operatorname{3}\)
\(\operatorname{0}\) \(\operatorname{0}\) \(\operatorname{1}\) \(\operatorname{2}\) \(\operatorname{3}\)
\(\operatorname{1}\) \(\operatorname{1}\) \(\operatorname{2}\) \(\operatorname{3}\) \(\operatorname{0}\)
\(\operatorname{2}\) \(\operatorname{2}\) \(\operatorname{3}\) \(\operatorname{0}\) \(\operatorname{1}\)
\(\operatorname{3}\) \(\operatorname{3}\) \(\operatorname{0}\) \(\operatorname{1}\) \(\operatorname{2}\)


\(\text{Table 35.2}~~\text{Regular Representation of the Group}~Z_4(+)\)
\(\text{Element}\!\) \(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(\operatorname{0}\) \(\{\!\) \((\operatorname{0}, \operatorname{0}),\) \((\operatorname{1}, \operatorname{1}),\) \((\operatorname{2}, \operatorname{2}),\) \((\operatorname{3}, \operatorname{3})\) \(\}\!\)
\(\operatorname{1}\) \(\{\!\) \((\operatorname{0}, \operatorname{1}),\) \((\operatorname{1}, \operatorname{2}),\) \((\operatorname{2}, \operatorname{3}),\) \((\operatorname{3}, \operatorname{0})\) \(\}\!\)
\(\operatorname{2}\) \(\{\!\) \((\operatorname{0}, \operatorname{2}),\) \((\operatorname{1}, \operatorname{3}),\) \((\operatorname{2}, \operatorname{0}),\) \((\operatorname{3}, \operatorname{1})\) \(\}\!\)
\(\operatorname{3}\) \(\{\!\) \((\operatorname{0}, \operatorname{3}),\) \((\operatorname{1}, \operatorname{0}),\) \((\operatorname{2}, \operatorname{1}),\) \((\operatorname{3}, \operatorname{2})\) \(\}\!\)


Sign Relations

Completed Work


\(\text{Table 1.} ~~ \text{Sign Relation of Interpreter A}\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)

\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)

\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)

\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)

\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)

\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)


\(\text{Table 2.} ~~ \text{Sign Relation of Interpreter B}\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)

\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)

\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)

\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)

\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)

\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)


\(\text{Table 36.} ~~ \text{Semantics for Higher Order Signs}\!\)
\(\text{Object Denoted}\!\) \(\text{Equivalent Signs}\!\)

\(\begin{matrix} \text{A} \\ \text{B} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} & = & {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\langle} \text{B} {}^{\rangle} & = & {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \end{matrix}\)

\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)

\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{A} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{A} {}^{\rangle\prime\prime} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{B} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{B} {}^{\rangle\prime\prime} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{i} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{i} {}^{\rangle\prime\prime} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{u} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{u} {}^{\rangle\prime\prime} \end{matrix}\)


\(\text{Table 37.} ~~ \text{Sign Relation Containing a Higher Order Sign}\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \text{s} \end{matrix}\)

\(\begin{matrix} \text{s} \\[2pt] \ldots \\[2pt] \text{t} \end{matrix}\)

\(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \ldots \end{matrix}\)


\(\text{Table 38.} ~~ \text{Sign Relation for a Succession of Higher Order Signs (1)}\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} x \\[2pt] {}^{\langle} x {}^{\rangle} \\[2pt] {}^{\langle\langle} x {}^{\rangle\rangle} \\[2pt] \ldots \end{matrix}\)

\(\begin{matrix} {}^{\langle} x {}^{\rangle} \\[2pt] {}^{\langle\langle} x {}^{\rangle\rangle} \\[2pt] {}^{\langle\langle\langle} x {}^{\rangle\rangle\rangle} \\[2pt] \ldots \end{matrix}\)

\(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \ldots \\[2pt] \ldots \end{matrix}\)


\(\text{Table 39.} ~~ \text{Sign Relation for a Succession of Higher Order Signs (2)}\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} x \\[2pt] s_1 \\[2pt] s_2 \\[2pt] \ldots \end{matrix}\)

\(\begin{matrix} s_1 \\[2pt] s_2 \\[2pt] s_3 \\[2pt] \ldots \end{matrix}\)

\(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \ldots \\[2pt] \ldots \end{matrix}\)


\(\text{Table 40.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(A)\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)


\(\text{Table 41.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(B)\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)


\(\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(A)\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\)


\(\text{Table 43.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(B)\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\)


Current Work


\(\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(A)\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)


\(\text{Table 45.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(B)\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) 


Table 44.  Higher Import Sign Relation HI1(A)
	Object	Sign	Interpretant
	A	<A>	<A>
	A	<A>	<i>
	A	<i>	<A>
	A	<i>	<i>
	B	<B>	<B>
	B	<B>	<u>
	B	<u>	<B>
	B	<u>	<u>
	<A, <A>, <A>>	<A>	<A>
	<A, <A>, <i>>	<A>	<A>
	<A, <i>, <A>>	<A>	<A>
	<A, <i>, <i>>	<A>	<A>
	<B, <B>, <B>>	<A>	<A>
	<B, <B>, <u>>	<A>	<A>
	<B, <u>, <B>>	<A>	<A>
	<B, <u>, <u>>	<A>	<A>
	<A, <A>, <A>>	<B>	<B>
	<A, <A>, <u>>	<B>	<B>
	<A, <u>, <A>>	<B>	<B>
	<A, <u>, <u>>	<B>	<B>
	<B, <B>, <B>>	<B>	<B>
	<B, <B>, <i>>	<B>	<B>
	<B, <i>, <B>>	<B>	<B>
	<B, <i>, <i>>	<B>	<B>

Table 45.  Higher Import Sign Relation HI1(B)
	Object	Sign	Interpretant
	A	<A>	<A>
	A	<A>	<u>
	A	<u>	<A>
	A	<u>	<u>
	B	<B>	<B>
	B	<B>	<i>
	B	<i>	<B>
	B	<i>	<i>
	<A, <A>, <A>>	<A>	<A>
	<A, <A>, <i>>	<A>	<A>
	<A, <i>, <A>>	<A>	<A>
	<A, <i>, <i>>	<A>	<A>
	<B, <B>, <B>>	<A>	<A>
	<B, <B>, <u>>	<A>	<A>
	<B, <u>, <B>>	<A>	<A>
	<B, <u>, <u>>	<A>	<A>
	<A, <A>, <A>>	<B>	<B>
	<A, <A>, <u>>	<B>	<B>
	<A, <u>, <A>>	<B>	<B>
	<A, <u>, <u>>	<B>	<B>
	<B, <B>, <B>>	<B>	<B>
	<B, <B>, <i>>	<B>	<B>
	<B, <i>, <B>>	<B>	<B>
	<B, <i>, <i>>	<B>	<B>
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