Shape

183 Shapes in Algebras and Algebras in Rows
is defined.)
I’ve described maximal elements in terms of embedding and boundaries because
it’s a cinch. But really, embedding is enough. The intuitive idea is this—basic elements
of the same kind are maximal with respect to one another when they’re separated
by gaps. To be exact, every basic element that overlaps any two in a shape has a basic
element embedded in it that overlaps neither. It’s in the gap. This is clear for the lines
Any line that overlaps both has the line
embedded in it that closes the gap between them. In fact, any line embedded in this
line overlaps neither of my original lines. But notice that maximal elements can also
touch. First, when they’re not embedded in a common element, as for these three
lines
but then, even if they are embedded in a common element, as for these squares
when boundary elements are discrete. Lines form boundaries of planes, while points
form boundaries of lines.
Maximal elements stand the customary atomic (analytic) view of things on its
head. Every shape is defined crudely with discrete elements that are numerically distinct,
but with no implied granularity or structure. The list of maximal elements contains
the smallest number of biggest things that combine to form the shape, instead of
the biggest number of smallest things that do. Of course, the atomic view is no more
than an approximation for shapes without points. Then every basic element has indefinitely
many others embedded in it. And so, maximal elements don’t behave like constituents.
Otherwise, both approaches are the same—as for points. The part relation
shows why.