Interval Dismantlable Lattices

A finite lattice is interval dismantlable if it can be partitioned into an ideal and a filter, each of which can be partitioned into an ideal and a filter, etc., until you reach 1-element lattices. In this note, we find a quasi-equational basis for the pseudoquasivariety of interval dismantlable lattices, and show that there are infinitely many minimal interval non-dismantlable lattices. Define an interval dismantling of a lattice to be a partition of the lattice into two nonempty, complementary sublattices where one is an ideal and the other a filter. A finite lattice is said to be interval dismantlable if it can be reduced to 1-element lattices by successive interval dismantlings. In order to work with these lattices, we note that the following are equivalent for a finite lattice L: (1) L = I ∪̇F for some disjoint proper ideal I and filter F . (2) L contains a nonzero join prime element. (3) L contains a non-one meet prime element. (4) There is a surjective homomorphism h : L→ 2. (5) Some generating set X for L can be split into two disjoint nonempty subsets, X = Y ∪̇Z, such that ∧ Y ∨ Z. (6) Every generating set X for L can be split into two disjoint nonempty subsets, X = Y ∪̇Z, such that ∧ Y ∨ Z. So if a lattice L contains no join prime element, then it is interval nondismantlable. If L contains no join prime element, but every proper sublattice does, then it is minimally interval non-dismantlable. If L contains an interval non-dismantlable sublattice, then L is interval nondismantlable. Note that it follows from (2) and (3) that every finite meet semidistributive or join semidistributive lattice is interval dismantlable. The Date: January 19, 2017. 2010 Mathematics Subject Classification. 06B05, 08C15.