Some varieties of semidistributive lattices. (English) Zbl 0572.06006
Universal algebra and lattice theory, Proc. Conf., Charleston/S.C. 1984, Lect. Notes Math. 1149, 198-223 (1985).
[For the entire collection see Zbl 0563.00005.]
It was shown by Jónsson and Rival that the lattice variety \(V(N_ 5)\) of the pentagon \(N_ 5\) has exactly 16 covers, 15 of which are generated by one of the finite nonmodular subdirectly irreducible lattices called \(L_ 1,...,L_{15}\). The lattice \(L_{11}\) is the dual of \(L_{12}\). The author’s result for \(V(L_{12})\) applies to \(V(L_{11})\) too: There are sequences of splitting lattices \(L^ n_{12}\), \(G^ n\), \(n\in \omega\) such that the varieties \(V(L^ 1_{12})\) and V(G) are the only join irreducible varieties covering \(V(L_{12})\) (Rose) and above each of these there is an ascending chain \(V(L^ n_{12})\) or \(V(G^ n)\) of varieties each of which is the only join irreducible variety covering the preceding one.
It was shown by Jónsson and Rival that the lattice variety \(V(N_ 5)\) of the pentagon \(N_ 5\) has exactly 16 covers, 15 of which are generated by one of the finite nonmodular subdirectly irreducible lattices called \(L_ 1,...,L_{15}\). The lattice \(L_{11}\) is the dual of \(L_{12}\). The author’s result for \(V(L_{12})\) applies to \(V(L_{11})\) too: There are sequences of splitting lattices \(L^ n_{12}\), \(G^ n\), \(n\in \omega\) such that the varieties \(V(L^ 1_{12})\) and V(G) are the only join irreducible varieties covering \(V(L_{12})\) (Rose) and above each of these there is an ascending chain \(V(L^ n_{12})\) or \(V(G^ n)\) of varieties each of which is the only join irreducible variety covering the preceding one.
Reviewer: G.Kalmbach
