The free \({\mathfrak m}\)-lattice on the poset H (Q761476)

The nonexistence of free complete lattices on arbitrary sets suggests a weakening of the concept of completeness. Let \({\mathfrak m}\) be a fixed infinite regular cardinal. A lattice L is called \({\mathfrak m}\)-lattice if in L all subsets X with \(0<| X| <{\mathfrak m}\) have meets and joins. Although \({\mathfrak m}\)-lattices have infinitary operations, there is a set (not a class) that can simultaneously index all the join and meet operations of every \({\mathfrak m}\)-lattice. A particular consequence is the existence, for any poset P, of \(F_{{\mathfrak m}}(P)\), the free \({\mathfrak m}\)-lattice on P (i.e. the unique \({\mathfrak m}\)-lattice for which every isotone map from P to any \({\mathfrak m}\)-lattice L has a unique extension to an \({\mathfrak m}\)-homomorphism from \(F_{{\mathfrak m}}(P)\) to L). Let \(H=\{a_ 0,a_ 1,a_ 2,b_ 0,b_ 1,b_ 2\}\) be the poset defined by \(a_ 0<a_ 2<a_ 1\), \(b_ 0<b_ 2<b_ 1\), \(a_ 0<b_ 1\) and \(b_ 0<a_ 1\). Continuing work of Rival and Wille (who accomplished the same for \({\mathfrak m}=\aleph_ 0)\) the paper gives the construction of the free \({\mathfrak m}\)-lattice on H, denoted by D(\({\mathfrak m})-\{\gamma,\gamma '\}\). In the sequel, a characterization of \(F_{{\mathfrak m}}(P)\) (P an arbitrary poset) due to Crawley and Dean is used.