Lattices of crosscuts
Let P be a finite poset. Then an antichain in P meets every maximal chain in P at most once, a cutset of P meets every maximal chain in P at least once, and a crosscut of P meets every maximal chain in P exactly once. In 1960, \textit{R. P. Dilworth} [Proc. Symp. Appl. Math. 10, 85-90 (1960; Zbl 0096.006)] proved that the maximum-sized antichains in P always form a distributive lattice (when ordered in a natural way) and recently \textit{K. M. Koh} [Algebra Univers. 17, 73-86 (1983; Zbl 0524.06003)] proved that every finite distributive lattice is isomorphic to the lattice of maximum-sized antichains of a suitable finite poset. Meanwhile, in 1972 \textit{F. Escalante} [Abh. Math. Sem. Univ. Hamb. 38, 199-220 (1972; Zbl 0239.05131)] proved that the minimum-sized cutsets of a finite poset also form a distributive lattice and he showed that every finite distributive lattice could be represented in this way. The purpose of the present paper is to obtain similar results for crosscuts. Crosscuts of arbitrary posets are defined and shown to form distributive lattices. Every finite distributive lattice is proved to be representable as the lattice of crosscuts of some finite poset. This representation is extended so as to include those of Escalante and Koh, as far as is possible. Essential use is made of some ideas of Escalante and Gallai.
scientific article; zbMATH DE number 3880764
scientific article