Disjointness conditions in free products of lattices
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.
scientific article; zbMATH DE number 3557837
scientific article