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===Commentary Note 9.5===

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~~<pre>~~

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Peirce's comma operation, in its application to an absolute term, is tantamount to the representation of that term's denotation as an idempotent transformation, which is commonly represented as a diagonal matrix. This is why I call it the "diagonal extension".

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Peirce's comma operation, in its application to an absolute term,

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is tantamount to the representation of that term's denotation as

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an idempotent transformation, which is commonly represented as a

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diagonal matrix. This is why I call it the "diagonal extension".

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An idempotent element x is given by the abstract condition that xx = x,

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An idempotent element ''x'' is given by the abstract condition that ''xx'' = ''x'', but we commonly encounter such elements in more concrete circumstances, acting as operators or transformations on other sets or spaces, and in that action they will often be represented as matrices of coefficients.

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but we commonly encounter such elements in more concrete circumstances,

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acting as operators or transformations on other sets or spaces, and in

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that action they will often be represented as matrices of coefficients.

Let's see how all of this looks from the graphical and matrical perspectives.

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Absolute terms:

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1 = "anybody" = B +, C +, D +, E +, I +, J +, O

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:{| cellpadding="4"

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| 1

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| = || "anybody"

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| = || B +, C +, D +, E +, I +, J +, O

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|-

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| m

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| = || "man"

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| = || C +, I +, J +, O

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|-

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| n

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| = || "noble"

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| = || C +, D +, O

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|-

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| w

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| = || "woman"

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| = || B +, D +, E

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|}

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~~m = "man" = C +, I +, J +, O~~

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Previously, we represented absolute terms as column vectors. The above four terms are given by the columns of this table:

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~~n = "noble" = C +, D +, O~~

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~~w = "woman" = B +, D +, E~~

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Previously, we represented absolute terms as column vectors.

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The above four terms are given by the columns of this table:

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<pre>

| 1 m n w

---o---------

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J | 1 1 0 0

O | 1 1 1 0

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</pre>

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One way to represent sets in the bigraph picture

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One way to represent sets in the bigraph picture is simply to mark the nodes in some way, like so:

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is simply to mark the nodes in some way, like so:

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<pre>

B C D E I J O

1 + + + + + + +

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B C D E I J O

w + o + + o o o

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</pre>

Diagonal extensions of the absolute terms:

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1, = "anybody that is ---" = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O

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:{| cellpadding="4"

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| 1,

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m, = "man that is ---" = C:C +, I:I +, J:J +, O:O

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| = || "anybody that is ---"

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| = || B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O

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n, = "noble that is ---" = C:C +, D:D +, O:O

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|-

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| m,

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w, = "woman that is ---" = B:B +, D:D +, E:E

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| = || "man that is ---"

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| = || C:C +, I:I +, J:J +, O:O

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|-

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| n,

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| = || "noble that is ---"

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| = || C:C +, D:D +, O:O

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|-

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| w,

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| = || "woman that is ---"

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| = || B:B +, D:D +, E:E

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|}

Naturally enough, the diagonal extensions are represented by diagonal matrices:

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<pre>

!1!| B C D E I J O

---o---------------

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J | 0 0 0 0 0 1 0

O | 0 0 0 0 0 0 1

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</pre>

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<pre>

!m!| B C D E I J O

---o---------------

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J | 0 0 0 0 0 1 0

O | 0 0 0 0 0 0 1

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</pre>

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<pre>

!n!| B C D E I J O

---o---------------

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J | 0 0 0 0 0 0 0

O | 0 0 0 0 0 0 1

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</pre>

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<pre>

!w!| B C D E I J O

---o---------------

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J | 0 0 0 0 0 0 0

O | 0 0 0 0 0 0 0

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</pre>

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Cast into the bigraph picture of 2-adic relations,

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Cast into the bigraph picture of 2-adic relations, the diagonal extension of an absolute term takes on a very distinctive sort of "straight-laced" character:

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the diagonal extension of an absolute term takes on

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a very distinctive sort of "straight-laced" character:

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<pre>

B C D E I J O

u o o o o o o o

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u o o o o o o o

B C D E I J O

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</pre>

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<pre>

B C D E I J O

u o o o o o o o

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u o o o o o o o

B C D E I J O

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</pre>

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<pre>

B C D E I J O

u o o o o o o o

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u o o o o o o o

B C D E I J O

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</pre>

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<pre>

B C D E I J O

u o o o o o o o

Jon Awbrey

12,080

edits