Narrow Results

For a partially ordered set P, let Co(P) denote the lattice of all order-convex subsets of P. For a positive integer n, we denote by (resp., SUB(n)) the class of all lattices that can be embedded into a lattice of the form

(1) Both classes and SUB(n), for any positive integer n, are locally finite, finitely based varieties of lattices, and we find finite equational bases of these varieties.

(2) The variety is the quasivariety join of all the varieties SUB(n), for 1≤n<ω, and it has only countably many subvarieties. We classify these varieties, together with all the finite subdirectly irreducible members of .

(3) Every finite subdirectly irreducible member of is projective within , and every subquasivariety of is a variety.

For a closure space (P, φ) with φ(ø) = ø, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P, φ), extending the poset Clop(P, φ) of all clopen subsets. If (P, φ) is a finite convex geometry, then Reg(P, φ) is pseudocomplemented. The Dedekind–MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg(P, φ) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces,

• Reg(P, φ) satisfies an infinite collection of stronger and stronger quasi-identities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity.

• If Reg(P, φ) is semidistributive, then it is a bounded homomorphic image of a free lattice.

• Clop(P, φ) is a lattice if and only if every regular closed set is clopen.

The extended permutohedron R(G) on a graph G and the extended permutohedron Reg S on a join-semilattice S, are both defined as lattices of regular closed sets of suitable closure spaces. While the lattice of all regular closed sets is, in the semilattice context, always the Dedekind–MacNeille completion of the poset of clopen sets, this does not always hold in the graph context, although it always does so for finite block graphs and for cycles. Furthermore, both R(G) and Reg S are bounded homomorphic images of free lattices.

For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation D_L on J(L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form:

Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators Δ(p), for p ∈ J(L), and relations

As a consequence of this, we obtain the following results:

Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the axiom

Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A □ B of A and B is join-semidistributive, and the following isomorphism holds: