Shape

191 Shapes in Algebras and Algebras in Rows
or have coincident elements in common
The maximal elements in a shape formed in a difference are not defined automatically.
Some work is needed to get them, just as it is to get the maximal elements in
a sum. An algorithm for lines is framed in terms of embedding in table 5. The rules
are independent, and they exhaust the ways any two lines interact when a segment
Table 5
Rules for Maximal Lines in Differences
Start with any two sets of maximal lines that are ordered by their endpoints. Let l be a line in the
first set and lA be a line in the second. Then, the first set is changed recursively according to three
rules that may be applied in any order until no rule can be applied.
(1) If l is embedded in lA
then remove l from the set.
(2) If lA is properly embedded in l, so that no endpoint is shared
then replace l with the lines lB and lBA fixed by the two leftmost and the two rightmost endpoints
of l and lA. Alternatively, if lA is properly embedded in l, and there is a common endpoint
then replace l with the line lB fixed by the remaining endpoints of l and lA.
(3) If l and lA overlap but neither is embedded in the other
then replace l with the line lB fixed either by the two leftmost or by the two rightmost endpoints
of l and lA, so that lB isn’t embedded in lA.