Cardinality questions concerning semilattices of finite breadth
Let A be a set and \({\mathcal F}(A)\) the finite subsets of A ordered by inclusion. R. Haydon has raised the question: Does there exist a cofinal family \({\mathcal F}_ 0\) (i.e. every G in \({\mathcal F}(A)\) is a subset of some F in \({\mathcal F}_ 0)\) which is closed under finite intersection and is such that for every F in \({\mathcal F}_ 0\) the collection of proper subsets of F which are in \({\mathcal F}_ 0\) has at most two maximal members? In this paper the author reformulates this question about \({\mathcal F}(A)\) in the setting of lattices, and provides a rather interesting answer. The author shows that if L is a lattice in which every element has only finitely many predecessors and not more than k immediate predecessors, for some positive integer k, then L has cardinality \(| L| \leq \aleph_{k-1}\). There are still other questions that remain open. For example, for the lattices mentioned above is \(| L| \leq \aleph_{k-1}\) possible for \(k>2\), or do we always have \(| L| \leq \aleph_ 1?\)
scientific article; zbMATH DE number 3845643