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We introduce a new class of poset edge labelings for locally finite latticeswhich we call $SB$-labelings. We prove for finite lattices which admit an$SB$-labeling that each open interval has the homotopy type of a ball or of asphere of some dimension. Natural examples include the weak order, the Tamarilattice, and the finite distributive lattices.

SB-labelings and posets with each interval homotopy equivalent to a sphere or a ball