Products of lattice varieties
The concept of variety product originated in \textit{H. Neumann}'s work on ''Varieties of groups'' [Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37 (1967; Zbl 0251.20001)] and was generalized to universal algebra by \textit{A. I. Mal'cev} [Sib. Mat. Zh. 8, 346-365 (1967; Zbl 0228.08007); The metamathematics of algebraic systems (1971; Zbl 0231.02002)]. Given two classes of lattices \({\mathcal V}\) and \({\mathcal W}\), their product \({\mathcal V}\circ {\mathcal W}\) consists of all lattices L which admit a congruence \(\Theta\) such that all congruence classes (as lattices with the induced order) belong to \({\mathcal V}\), while L/\(\Theta\) is in \({\mathcal W}\). The product operation preserves closedness under isomorphic copies, sublattices, products and ultraproducts, but unfortunately not closedness under homomorphic images. E.g., if \({\mathcal V}\) is a nondistributive variety generated by a finite lattice and \({\mathcal D}\) consists of all distributive lattices, then \({\mathcal V}\circ {\mathcal D}\) is not a variety (Theorem 1). This contrasts with the situation for groups, where products of varieties are again varieties. The main result of the paper is the embedding theorem (Theorem 2). If the class \({\mathcal V}\) is closed under formation of ideal- and filter-lattices, then every object L of \({\mathcal V}\circ {\mathcal W}\) by virtue of a congruence \(\Theta\) can be embedded into a lattice \(\hat L,\) which belongs to \({\mathcal V}\circ {\mathcal W}\) by virtue of an extension \({\hat \Theta}\) of \(\Theta\), and all \({\hat \Theta}\)-classes of \(\hat L\) are complete lattices. Moreover, L/\(\Theta\) \(\cong \hat L/{\hat \Theta}\), and every congruence of L extends to \(\hat L.\) This leads to a sufficient condition for the product of two lattice varieties to be a variety. The authors then provide a construction scheme, which for varieties \({\mathcal V}\) and \({\mathcal W}\) produces all the lattices \(\hat L\) for the objects L of \({\mathcal V}\circ {\mathcal W}\) from building blocks in \({\mathcal V}\) and \({\mathcal W}.\) Motivated by a result of \textit{A. Day} [Algebra Univers. 7, 163-169 (1977; Zbl 0381.06010)] that the variety of all lattices is the only lattice variety containing every iterated product of \({\mathcal D}\) with itself, the authors introduce a notion of dimension for lattice varieties. With the help of a dimension formula for products they proceed to show that every nontrivial lattice variety can be written as a product of \(\circ\)- irreducible ones (which by definition are not products of nontrivial varieties). \{Reviewer's remark: The most serious of several misprints occurs in condition (iii) of Theorem 2. It should read: every \({\hat \Theta}\)-class of \(\hat L\) is a complete lattice. Moreover, the dimension formula for Lemma 9 requires the hypothesis that \({\mathcal V}\circ {\mathcal W}\) is a variety, or an extension of the dimension concept for prevarieties.\}
scientific article; zbMATH DE number 3941602
scientific article