Motivated by some examples
from the study of axiomatic convexity, we define a class of objects (in real algebras
with 1) whose algebraic properties mimic those of the unit interval. These objects,
called intervals, have quite a bit of structure in themselves. In particular, in a Banach
algebra a compact interval must be finite dimensional. Even more striking is the main
result which shows that any interval satisfying a very modest boundedness condition
is commutative and can be represented by continuous functions from a compact
Hausdorff space into the unit interval. This leads to a number of corollaries in
analysis and topology.
