Shape

187 Shapes in Algebras and Algebras in Rows
Table 4
Reduction Rules for Maximal Lines
Assume that lines are ordered by their endpoints, and let l and lA be any two of these lines. Then,
a set of lines is changed into a set of maximal lines according to three reduction rules. The rules
are used recursively in any order until no rule can be applied.
(1) If l is embedded in lA, either like this
so that there’s a common endpoint, or like this
so that there isn’t, then remove l from the arrangement.
(2) If l and lA overlap but neither is embedded in the other
then replace both lines with the line lB fixed by the leftmost endpoint of l and the rightmost end
point of lA. This is the longest line with an endpoint of l and an endpoint of lA.
(3) If l and lA are collinear and discrete, and share an endpoint
then replace both lines with the line lB fixed by the remaining endpoints of l and lA.
inconsistency—until maximal lines are finally defined. If I begin with the maximal
lines in two shapes, I get the maximal lines in their sum. (It’s worth saying, as well,
that the equivalence relation I defined to catalogue sets of basic elements can be given
in terms of these reduction rules—call the three of them working together R. The sets S
and T define the same shape if and only if RðSÞ ¼RðTÞ. Be careful, though. The part
relation isn’t the subset relation. The set S can be included in the set T—then RðSÞ is
part of RðTÞ—without RðSÞ being a subset of RðTÞ. The subset relation requires identity—embedding
won’t do.)
The four lines in this scheme
are reduced to a single maximal line after each of the rules in table 4 is applied once,
perhaps in this sequence