  The jeu de taquin process produced a standard Young tableau from a skew
standard Young tableau by shifting its entries to the northwest. We generalize
this process to posets: certain partial numberings of any poset are shifted
upward. A poset is said to have the jeu de taquin property if the numberings
resulting from this process do not depend upon certain choices made during the
process. Young diagrams are the posets which underlie standard Young tableaux.
These posets have the jeu de taquin property. d-Complete posets are posets
which satisfy certain local structual conditions. They are mutual
generalizations of Young diagrams, shifted Young diagrams, and rooted trees. We
prove that all d-complete posets have the jeu de taquin property. The proof
shows that each d-complete poset actually has the stronger "simultaneous"
property; this may lead to an algebraic understanding of the main result. A
partial converse is stated: "Non-overlapping" simultaneous posets are
d-complete.
