The subvariety lattice of the variety of distributive doublep-algebras
Wiesław Dziobiak · Bulletin of the Australian Mathematical Society · 1985
LetLdenote the subvariety lattice of the variety of distributive doublep-algebras, that is, the lattice whose universe consists of all varieties of distributive doublep-algebras and whose ordering is the inclusion relation. We prove in this paper that each proper filter inLis uncountable. Moreover, we prove that except for the trivial variety (the zero inL) and the variety of Boolean algebras (the unique atom inL) every other element ofL, generated by a finite algebra, has infinitely many covers inL, among which at least one is not generated by any finite algebra. The former result strengthens a result of Urquhart who showed that the latticeLis uncountable. On the other hand, both of our results indicate a high complexity of the latticeLat least in comparison with the subvariety lattice of the variety of distributivep-algebras, since a result of Lee shows that the latter lattice forms a chain of type ω + 1 and every cover in it of the variety generated by a finite algebra is itself generated by a finite algebra.