Changes

===Commentary Note 11.15===

===Commentary Note 11.15===

I'm going to elaborate a little further on the subject of arrows, morphisms, or structure-preserving maps, as a modest amount of extra work at this point will repay ample dividends when it comes time to revisit Peirce's "number of" function on logical terms.

I'm going to elaborate a little further on the subject

 

of arrows, morphisms, or structure-preserving maps, as

a modest amount of extra work at this point will repay

 

ample dividends when it comes time to revisit Peirce's

"number of" function on logical terms.

 

 

 

Real number addition and real number multiplication (suitably restricted) are examples of group operations.  If we write the sign of each operation in braces as a name for the 3-adic relation that constitutes or defines the corresponding group, then we have the following set-up:

is just the structure that we all know and love as a 3-adic relation.

Very typically, it will be the type of 3-adic relation that defines

the type of 2-ary operation that obeys the rules of a mathematical

structure that is known as a "group", that is, a structure that

satisfies the axioms for closure, associativity, identities,

For example, in the previous case of the logarithm map J, we have the data:

: ''J'' : {+} &larr; {<math>\cdot</math>}

| J : R <- R (properly restricted)

: {+} &sube; '''R''' &times; '''R''' &times; '''R'''

| K : R <- R x R, where K(r, s) = r + s

| L : R <- R x R, where L(u, v) = u . v

: {<math>\cdot \;</math>} &sube; '''R''' &times; '''R''' &times; '''R'''

the corresponding group, then we have the following set-up:

| J : {+} <- {.}

In many cases, one finds that both groups are written with the same sign of operation, typically "<math>\cdot</math>", "+", "*", or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used. In such a setting, our chiasmatic theme may run a bit like these two variants:

| {+} c R x R x R

| {.} c R x R x R

In many cases, one finds that both groups are written with the same

: The image of the sum is the sum of the images.

sign of operation, typically ".", "+", "*", or simple concatenation,

: The image of the product is the product of the images.

| The image of the product is the product of the images.

Figure 22 presents a generic picture for groups G and H.

Figure 22 presents a generic picture for groups ''G'' and ''H''.

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

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

|                                                          |

|                                                          |

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

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

Figure 22.  Group Homomorphism J : G <- H

Figure 22.  Group Homomorphism J : G <- H

In a setting where both groups are written with a plus sign,

In a setting where both groups are written with a plus sign, perhaps even constituting the very same group, the defining formula of a morphism, ''J''(''L''(''u'',&nbsp;''v'')) = ''K''(''Ju'',&nbsp;''Jv''), takes on the shape ''J''(''u''&nbsp;+&nbsp;''v'') = ''Ju''&nbsp;+&nbsp;''Jv'', which looks very analogous to the'distributive multiplication of a sum (''u''&nbsp;+&nbsp;''v'') by a factor ''J''. Hence another popular name for a morphism:  a "linear" map.

perhaps even constituting the very same group, the defining

formula of a morphism, J(L(u, v)) = K(Ju, Jv), takes on the

shape J(u + v) = Ju + Jv, which looks very analogous to the

distributive multiplication of a sum (u + v) by a factor J.

Hence another popular name for a morphism:  a "linear" map.

===Commentary Note 11.16===

===Commentary Note 11.16===