Chain conditions in free products of lattices (Q1239173)

A poset satisfies the \(\mathfrak m\)-chain condition (for infinite cardinal \(\mathfrak m\)) provided all chains have cardinality \(<\mathfrak m\). For the variety of distributive lattices \(\mathcal D\), \textit{G. Grätzer} and \textit{H. Lakser} [Trans. Am. Math. Soc. 144, 301--312 (1969; Zbl 0194.32504)] show that the distributive free product preserves the \(\mathfrak m\)-chain condition for uncountable regular \(\mathfrak m\). It is shown here that for any variety \(\mathcal V\) of lattices and uncountable regular cardinal \(\mathfrak m\), the completely \(\mathcal V\)-free lattice construction over a poset preserves the \(\mathfrak m\)-chain condition. Further, for the varieties \(\mathcal L\) (of all lattices) and \(\mathcal D\), the \(V\)-free product of a family \(L_i\) (\(i\in I\)) of lattices, \(L_i \in \mathcal V\), is order-isomorphic to a subposet of the completely \(\mathcal V\)-free lattice over the poset \(\dot{\bigcup} (L_i\mid i\in I)\). Consequently, the free product in the variety \(\mathcal L\) preserves the \(\mathfrak m\)-chain condition for uncountable regular \(\mathfrak m\).