  We study two natural preorders on the set of tripotents in a JB$^*$-triple
defined in terms of their Peirce decomposition and weaker than the standard
partial order. We further introduce and investigate the notion of finiteness
for tripotents in JBW$^*$-triples which is a natural generalization of
finiteness for projections in von Neumann algebras. We analyze the preorders in
detail using the standard representation of JBW$^*$-triples. We also provide a
refined version of this representation - in particular a decomposition of any
JBW$^*$-triple into its finite and properly infinite parts. Since a
JBW$^*$-algebra is finite if and only if the extreme points of its unit ball
are just unitaries, our notion of finiteness differs from the concept of
modularity widely used in Jordan structures so far. The exact relationship of
these two notions is clarified in the last section.
