On partially ordered sets embeddable in a free lattice
For elements p, p' of a poset P, write \(p\sim p'\) provided \(x<p\) if and only if \(x<p'\) and dually. Theorem: P is embeddable in a free lattice if and only if P/\(\sim\) is so embeddable. A bipartite poset P\({\dot \cup}Q\) consists of two infinite antichains P, Q. A crown in P\({\dot \cup}Q\) consists of n-element subsets \(\{p_ 0,...,p_{n-1}\}\) and \(\{q_ 0,...,q_{n-1}\}\) of P and Q, respectively, such that \(p_ 0\leq q_ 0>p_ 1<q_ 1>...<q_{n-1}>p_ 0\) and \(p_ i\nleq q_ j\) for \(j\neq i,i-1(mod n)\). Theorem: If \(S=P{\dot \cup}Q\) is bipartite and no element is part of an infinite number of crowns, then FL(S), the lattice freely generated by S, is projective. Theorem: Suppose P\({\dot \cup}Q\) is embeddable in a free lattice. If P/\(\sim\) is finite, then \(| Q/\sim | \leq 2^{| P/\sim |}\); if P/\(\sim\) is infinite, then \(| Q/\sim | =| P/\sim |\). Theorem: Suppose that P\({\dot \cup}Q\) is embeddable in a free lattice, that \(\sim\) is equality, and that \(| P| =| Q| =\beta\), a regular uncountable cardinal. Then there exist \(P^*\subset P\), \(Q^*\subset Q\) such that \(| P^*| =| Q^*| =\beta\) and \(P^*\cup Q^*\) is an antichain.
scientific article; zbMATH DE number 3966141