  Although d-complete posets arose along the interface between algebraic
combinatorics and Lie theory, they are defined using only requirements on their
local structure. These posets are a mutual generalization of rooted trees,
shapes, and shifted shapes. They possess Stanley's hook product property for
their P-partition generating functions and Schutzenberger's well defined jeu de
taquin rectification property. The original definition of d-complete poset was
lengthy, but more succinct definitions were later developed. Here several
definitions are shown to be equivalent. The basic properties of d-complete
posets are summarized. Background and a partial bibliography for these posets
is given.
