Cardinality questions concerning semilattices of finite breadth
type: Item
instance of: scholarly article
title: Cardinality questions concerning semilattices of finite breadth
zbMATH Open document ID: 0533.06005
DOI: 10.1016/0012-365X(84)90129-8
published in: Discrete Mathematics
publication date: 1984-01-01T00:00:00Z
review text: 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 \(| L| \leq \aleph_ 1?\)
Mathematics Subject Classification ID: 06B25
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 06A12
zbMATH DE Number: 3845643
zbMATH Keywords: k-lattice
zbMATH Keywords: irredundant sequence
zbMATH Keywords: breadth
zbMATH Keywords: cofinal family
zbMATH Keywords: maximal members
zbMATH Keywords: predecessors
author: Seymour Z. Ditor
MaRDI profile type: Publication
cites work: Q5331549
cites work: On the structure of set-mappings
cites work: Sur une caractérisation des alephs
cites work: Q5551488
description: scientific article; zbMATH DE number 3845643