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A subset F of an ordered set X is a fibre of X if F intersects every maximal antichain of X. We find a lower bound on the function / (£>), the minimum fibre size in the distributive lattice D, in terms of the size of D. In particular, we prove that there is a constant c such that f(D)>c- In the process we show that minimum fibre size is a monotone property for a certain class of distributive lattices. This fact depends upon being able to split every maximal antichain of this class ofdoi:10.1017/s144678870000238x fatcat:q5ts52laz5cyxp2hv6vef6a5bq