Disjointness conditions in free products of lattices
type: Item
instance of: scholarly article
title: Disjointness conditions in free products of lattices
zbMATH Open document ID: 0358.06015
DOI: 10.1007/BF02485434
author: K. Appert
published in: Algebra Universalis
publication date: 1977-01-01T00:00:00Z
review text: Let \(L\) be a lattice and \(\mathfrak m\) be an infinite cardinal. \(A\) subset \(D\) of \(L\) is disjoint (quasidisjoint) if there is \(a \in L\) with \(x\not\leq a\) for all \(x\in D\) such that \(x\land y=a\) (\(x\land y\leq a\)) for any distinct \(x,y \in D\). \(L\) satisfies the \(\mathfrak m\)-disjointness (\(\mathfrak m\)-quasidisjointness) condition if every disjoint (quasidisjoint) subset of \(L\) has cardinality \(<\mathfrak m\). For the variety \(\mathcal L\) of all lattices, it is shown that the free product preserves the \(\mathfrak m\)-disjointness condition for any \(\mathfrak m\). Further, the free product preserves the same quasidisjointness conditions in \(\mathcal L\) and in the variety \(\mathcal D\) of distributive lattices. (The disjointness and quasidisjointness conditions coincide for \(\mathcal D\)). For \(\mathcal V=\mathcal D\) or \(\mathcal L\), the \(\mathfrak m\)-quasidisjointness condition is preserved by the \(\mathcal V\)-free product if \(\mathfrak m\) is weakly compact or singular (assuming the Generalized Continuum Hypothesis). For \(\mathcal V= \mathcal D\), this generalizes a result of \textit{H. Lakser} [in: Proc. Univ. Houston, Lattice Theory Conf., Houston 1973, 156--168 (1973; Zbl 0335.06006)]. Let \(\mathfrak m\) be a non weakly compact cardinal. Examples are given, under the (consistent) assumption that there exists an \(\mathfrak m\)-Souslin tree, for which the \(\mathfrak m\)-quasidisjointness condition is not preserved in \(\mathcal V\)-free products.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B20
zbMATH DE Number: 3557837
MaRDI profile type: Publication
cites work: Chain conditions in free products of lattices
cites work: Projective and injective distributive lattices
cites work: Free Lattices Generated by Partially Ordered Sets and Preserving Bounds
cites work: Partition relations for cardinal numbers
cites work: Distributive Sublattices of a Free Lattice
cites work: Free products of lattices
cites work: Lectures in set theory with particular emphasis on the method of forcing
cites work: The fine structure of the constructible hierarchy
cites work: Relatively free products of lattices
cites work: Q4101867
cites work: Q4080379
full work available at URL: https://doi.org/10.1007/bf02485434
OpenAlex ID: W2061285536
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description: scientific article; zbMATH DE number 3557837