Note on the order dimension of partially ordered sets

Abstract

Proof. Let 2 e be generated by chains Ki (is1) with III =m. By Zorn's Lemma, there are maximal chains/~i in 2 e with Ki~_Ki ( isI ) . For any a s P there exists a greatestf~ a in/s with fi"(a)= 0 since/s is maximal. Define binary relations 091 on P by aogib:.~fi">~fi b (a, bsP, i s I ) . Obviously, the o)i's are quasi-orders. Since the/~i's are maximal, the coi's are even (linear) orders. L e t f " s 2 e be the isotone mapping satisfying f ~ (x) = O.*~-x ~fi">~k~ for i s l , it follows that f a = = l,_)i~If~". NOW, we get that a ~f~ b in 2 P for all i s L Thus, the partial order of P is the intersection of all orders co t ( / e l ) ; i.e. P has order dimension ~ I,.] i~ i g~ for asP. Let b s P with f b >t t> [,.)i~I g~. Then b~o~a for all i s I ; hence b<<.a by assumption. Since every element f of 2 e is the infimum of e lementsf x, we get f " = ~.Ji~ i ga for aeP. Thus, 2 e is generated as complete lattice by the m chains Ki (iEI). Using the well-known fact that every finite distributive lattice D is antiisomorphic to the lattice 2 J(~ where J(D) is the partially ordered set of all join-irreducible elements ( # 0 ) of D (c.f. Birkhoff [1], Theorem IX.5) we get the following corollary.