Changes

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.

The "structure" that is being preserved by a structure-preserving map 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, and inverses.

The ''structure'' that is preserved by a structure-preserving map 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, and inverses.

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

For example, in the previous case of the logarithm map <math>J,\!</math> we have the data:

: ''J'' : '''R''' &larr; '''R''' (properly restricted)

 

: ''K'' : '''R''' &larr; '''R''' &times; '''R''', where ''K''(''r'', ''s'') = ''r'' + ''s''

 

J & : & \mathbb{R} & \leftarrow & \mathbb{R}

: ''L'' : '''R''' &larr; '''R''' &times; '''R''', where ''L''(''u'', ''v'') = ''u'' <math>\cdot</math> ''v''

& ~\text{(properly restricted)}

K & : & \mathbb{R} & \leftarrow & \mathbb{R} \times \mathbb{R}

& \text{where}~ K(r, s) = r + s

L & : & \mathbb{R} & \leftarrow & \mathbb{R} \times \mathbb{R}

& \text{where}~ L(u, v) = u \cdot v

\end{array}</math>

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:

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: