  Overlap coincidence is an equivalent criterion to pure discrete spectrum of
the dynamics of self affine tilings. In the case of one dimension, strong
coincidence on m letter irreducible substitution has been introduced in Dekking
(1978) and Arnoux and Ito (2001) which implies that the system is metrically
conjugate to a domain exchange. However being a domain exchange does not imply
the property of pure discrete spectrum of the tiling dynamics. The relation
between two coincidences has not been established completely. In this paper we
generalize strong coincidence to higher dimensions and show the implication
from overlap coincidence to the new strong coincidence when the associated
height group is trivial. Furthermore we introduce a new criterion simultaneous
coincidence and show the implication from overlap coincidence to the
simultaneous coincidence. The triviality of the height group is shown in Barge
and Kwapisz (2006) and Sing (2006) for 1 dimension irreducible Pisot
substitutions.
