algebraic representation of relation composition

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The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the introduction of coordinates, in other words, identifiable names for the objects that are related through the various forms of relations, 2-adic and 3-adic in the present case. Adding coordinates to the running Example produces the following Figure:

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Figure 7.  F as the Intersection of tau(G) and tau(H)

Thinking of relations in operational terms is facilitated by using a variant notation for tuples and sets of tuples, namely, the ordered pair $(x,y)$ is written $x : y$ , the ordered triple $(x,y,z)$ is written $x : y : z$ , and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like $a\mathrm{:}b+b\mathrm{:}c+c\mathrm{:}d$ and so on.

For example, translating the relations $F\subseteq X\times Y\times Z$ , $G\subseteq X\times Y$ , $H\subseteq Y\times Z$ into this notation produces the following summary of the data:

$\begin{array}[]{ccccccc}F&=&4:3:4&+&4:4:4&+&4:5:4\\ G&=&4:3&+&4:4&+&4:5\\ H&=&3:4&+&4:4&+&5:4\\ \end{array}$

As often happens with abstract notations for functions and relations, the type information, in this case, the fact that $G$ and $H$ live in different spaces, is left implicit in the context of use.

Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of 2-adic relations in terms of the medial projection of the intersection of their tacit extensions:

$G\circ H=\mathrm{proj}_{XZ}(\tau_{XY}^{Z}(G)\ \cap\ \tau_{YZ}^{X}(H)).$

Here is the big picture, with all of the pieces in place:

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Figure 8.  G o H  =  proj_XZ (tau(G) |^| tau(H))

All that remains is to check the following collection of data and derivations against the situation represented in Figure 8.

$\begin{array}[]{ccccccc}F&=&4:3:4&+&4:4:4&+&4:5:4\\ G&=&4:3&+&4:4&+&4:5\\ H&=&3:4&+&4:4&+&5:4\\ \end{array}$

$\begin{array}[]{lcc}G\circ H&=&(4\mbox{:}3+4\mbox{:}4+4\mbox{:}5)(3\mbox{:}4+4% \mbox{:}4+5\mbox{:}4)\\ &=&4\mbox{:}4\\ \end{array}$

$\begin{array}[]{lcc}\tau(G)&=&\tau_{XY}^{Z}(G)\\ &=&\sum_{z=1}^{7}(4\mbox{:}3\mbox{:}z+4\mbox{:}4\mbox{:}z+4\mbox{:}5\mbox{:}z)% \\ \end{array}$

$\begin{array}[]{lccccccc}\tau(G)&=&4\mbox{:}3\mbox{:}1&+&4\mbox{:}4\mbox{:}1&+% &4\mbox{:}5\mbox{:}1&+\\ &&4\mbox{:}3\mbox{:}2&+&4\mbox{:}4\mbox{:}2&+&4\mbox{:}5\mbox{:}2&+\\ &&4\mbox{:}3\mbox{:}3&+&4\mbox{:}4\mbox{:}3&+&4\mbox{:}5\mbox{:}3&+\\ &&4\mbox{:}3\mbox{:}4&+&4\mbox{:}4\mbox{:}4&+&4\mbox{:}5\mbox{:}4&+\\ &&4\mbox{:}3\mbox{:}5&+&4\mbox{:}4\mbox{:}5&+&4\mbox{:}5\mbox{:}5&+\\ &&4\mbox{:}3\mbox{:}6&+&4\mbox{:}4\mbox{:}6&+&4\mbox{:}5\mbox{:}6&+\\ &&4\mbox{:}3\mbox{:}7&+&4\mbox{:}4\mbox{:}7&+&4\mbox{:}5\mbox{:}7&\\ \end{array}$

$\begin{array}[]{lcc}\tau(H)&=&\tau_{YZ}^{X}(H)\\ &=&\sum_{x=1}^{7}(x\mbox{:}3\mbox{:}4+x\mbox{:}4\mbox{:}4+x\mbox{:}5\mbox{:}4)% \\ \end{array}$

$\begin{array}[]{lccccccc}\tau(H)&=&1\mbox{:}3\mbox{:}4&+&1\mbox{:}4\mbox{:}4&+% &1\mbox{:}5\mbox{:}4&+\\ &&2\mbox{:}3\mbox{:}4&+&2\mbox{:}4\mbox{:}4&+&2\mbox{:}5\mbox{:}4&+\\ &&3\mbox{:}3\mbox{:}4&+&3\mbox{:}4\mbox{:}4&+&3\mbox{:}5\mbox{:}4&+\\ &&4\mbox{:}3\mbox{:}4&+&4\mbox{:}4\mbox{:}4&+&4\mbox{:}5\mbox{:}4&+\\ &&5\mbox{:}3\mbox{:}4&+&5\mbox{:}4\mbox{:}4&+&5\mbox{:}5\mbox{:}4&+\\ &&6\mbox{:}3\mbox{:}4&+&6\mbox{:}4\mbox{:}4&+&6\mbox{:}5\mbox{:}4&+\\ &&7\mbox{:}3\mbox{:}4&+&7\mbox{:}4\mbox{:}4&+&7\mbox{:}5\mbox{:}4&\\ \end{array}$

$\begin{array}[]{cll}\tau(G)\cap\tau(H)&=&4\mbox{:}3\mbox{:}4+4\mbox{:}4\mbox{:% }4+4\mbox{:}5\mbox{:}4\\ G\circ H&=&\mbox{proj}_{XZ}(\tau(G)\cap\tau(H))\\ &=&\mbox{proj}_{XZ}(4\mbox{:}3\mbox{:}4+4\mbox{:}4\mbox{:}4+4\mbox{:}5\mbox{:}% 4)\\ &=&4:4\\ \end{array}$

Title	algebraic representation of relation composition
Canonical name	AlgebraicRepresentationOfRelationComposition
Date of creation	2013-03-22 17:50:25
Last modified on	2013-03-22 17:50:25
Owner	Jon Awbrey (15246)
Last modified by	Jon Awbrey (15246)
Numerical id	5
Author	Jon Awbrey (15246)
Entry type	Example
Classification	msc 68R01
Classification	msc 68P15
Classification	msc 08A02
Classification	msc 05C65
Classification	msc 05B30
Classification	msc 05B20
Classification	msc 03E20
Classification	msc 03B10
Related topic	GeometricRepresentationOfRelationComposition
Related topic	MatrixRepresentationOfRelationComposition
Related topic	GraphTheoreticRepresentationOfRelationComposition