Shape

185 Shapes in Algebras and Algebras in Rows
all. Making shapes out of basic elements is a sweet thing to do. Even so, the lattice
properties of shapes are usually framed in terms of Boolean algebras, and this is equally
telling.
The familiar Boolean operations are used to combine shapes in various ways in
the algebras U ij . Sum (join), product (meet), and difference will do, or equivalently,
symmetric difference and product. The part relation provides for these definitions
regardless of the value of i. Once again, embedding is the key idea that supports my
entire approach to shapes.
A unique sum is formed whenever two shapes are added together. Twin conditions
are met—(1) both shapes are parts of the sum, and (2) every part of the sum has
a part that’s part of one shape or the other. (To make this perfect, I should say that
every nonempty part of the sum has a nonempty part that’s part of one shape or the
other, because the empty shape is part of every shape. I’ll avoid this detail again when I
define difference.) With just condition 1, the sum can be too big, with condition 2, too
small, but with both conditions, it’s just right. Consider some examples for different
values of i. The shape
with eight points is formed when the shape
with five points is added to the shape
with six points. The same three points
occur in both shapes to account for the eight-point sum. Every part of the sum—
except, of course, the empty one—is a combination of points from either one or