Changes

==Note 25==

==Note 25==

Continuing with the example <math>pq : X \to \mathbb{B},</math> Figure&nbsp;25-1 shows the enlargement or shift map <math>\operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math> in the same style of differential field picture that we drew for the tacit extension <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>

Staying with the example pq : X -> B, Figure 25-1 shows

the enlargement or shift map E[pq] : EX -> B in the same

style of differential field picture that we drew for the

tacit extension !e![pq] : EX -> B.

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

{| align="center" cellspacing="10" style="text-align:center"

|                                                                     |

| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]

|   X                                                                |

|-

|           o-------------------o  o-------------------o            |

| <math>\text{Figure 25-1.  Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>

|          /                    \ /                     \          |

|-

|         /  P                    o                    Q  \         |

|

|        /                      / \                       \         |

<math>\begin{array}{rcccccc}

|        /                      /  \                       \       |

\operatorname{E}(pq)

|      /                      /    \                       \       |

|      /                      /      \                       \     |

|    /                      /        \                       \     |

& \cdot &

& \cdot &

|    |                      |  \   /  |                      |    |

\texttt{(} \operatorname{d}p \texttt{)}

|    |            (dp) dq  |    \ /    |  dp (dq)             |    |

\texttt{(} \operatorname{d}q \texttt{)}

\\[4pt]

& \cdot &

\texttt{(} q \texttt{)}

|    \                       \   |    /                      /    |

|      \                       \   |  /                      /      |

\texttt{(} \operatorname{d}p \texttt{)}

|      \                       \ |  /                      /      |

|        \                       \ | /                      /        |

|        \                       \|/                      /        |

|          \                       |                      /          |

|          \                     /|\                     /          |

& \cdot &

\texttt{~} \operatorname{d}p \texttt{~}

\texttt{(} \operatorname{d}q \texttt{)}

\\[4pt]

|                                                                     |

\texttt{(} p \texttt{)}

& \cdot &

\texttt{(} q \texttt{)}

|}

A very important conceptual transition has just occurred here,

A very important conceptual transition has just occurred here, almost tacitly, as it were.  Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields <math>\varepsilon f</math> and <math>\operatorname{E}f,</math> both of the type <math>\operatorname{E}X \to \mathbb{B},</math> is very useful, because it allows us to consider these fields as integral mathematical objects that can be operated on and combined in the ways that we usually associate with algebras.

almost tacitly, as it were.  Generally speaking, having a set

of mathematical objects of compatible types, in this case the

two differential fields !e!f and Ef, both of the type EX -> B,

is very useful, because it allows us to consider these fields

as integral mathematical objects that can be operated on and

combined in the ways that we usually associate with algebras.

In this case one notices that the tacit extension !e!f and the

In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\operatorname{E}f</math> are in a certain sense dual to each other.  The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\operatorname{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true.  The only arc they have in common is the no-change loop <math>\texttt{(} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{)}</math> at <math>pq.\!</math> If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\operatorname{D}(pq) = \varepsilon(pq) + \operatorname{E}(pq)</math> that are illustrated in Figure&nbsp;25-2.

enlargement Ef are in a certain sense dual to each other, with

!e!f indicating all of the arrows out of the region where f is

true, and with Ef indicating all of the arrows into the region

where f is true.  The only arc that they have in common is the

no-change loop (dp)(dq) at pq.  If we add the two sets of arcs

mod 2, then the common loop drops out, leaving the 6 arrows of

D[pq] = !e![pq] + E[pq] that are illustrated in Figure 25-2.

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

{| align="center" cellspacing="10" style="text-align:center"

|                                                                     |

| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]

|  X                                                                |

|-

|           o-------------------o  o-------------------o            |

| <math>\text{Figure 25-2.  Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>

|           /                    \ /                    \           |

|-

|          /  P                    o                    Q  \         |

|

|        /                      / \                       \         |

<math>\begin{array}{rcccccc}

|       /                      /  \                       \       |

\operatorname{D}(pq)

|    /                      /        \                      \     |

& \cdot &

|    |            (dp) dq  |          |  dp (dq)             |    |

\texttt{(} \operatorname{d}p \texttt{)}

\texttt{(} \operatorname{d}q \texttt{)}

|    \                       \   |    /                      /    |

|      \                       \   |  /                      /      |

\texttt{(} q \texttt{)}

|      \                       \ |  /                      /      |

& \cdot &

|        \                       \ | /                      /        |

\texttt{~}

|        \                       \|/                      /        |

\texttt{(} \operatorname{d}p \texttt{)}

|          \                       |                      /          |

\texttt{~} \operatorname{d}q \texttt{~}

|          \                     /|\                     /          |

\texttt{~}

\\[4pt]

\texttt{(} p \texttt{)}

Figure 25-2.  Difference Map D[pq] : EX -> B

\texttt{(} \operatorname{d}q \texttt{)}

 

the underlying universe X% = [p, q] to give the following expansion:

& + &

 

\texttt{(} p \texttt{)}

 

\texttt{(}q \texttt{)}

 

  p q . ((dp)(dq))

 

\texttt{~} \operatorname{d}q \texttt{~}

  +

 

\end{array}</math>

  p (q) . (dp) dq

 

 

  (p) q . dp (dq)

 

 

  (p)(q) . dp dq

</pre>

==Note 26==

==Note 26==