Derived Semidistributive Lattices

TL;DR

This paper characterizes semidistributive lattices through connected components of a specific poset, introduces new characterizations of join semidistributive and lower bounded lattices, and explores their structure with explicit examples.

Contribution

It provides new characterizations of finite join semidistributive and lower bounded lattices, and demonstrates how these properties influence the structure of derived lattices.

Findings

C(L,g) is a semidistributive lattice if L is semidistributive.

C(L,g) is a bounded lattice if L is bounded.

Explicit computations relate C(S_n,a) and C(T_n,a) to smaller permutohedra and associahedra.

Abstract

For L a finite lattice, let C(L) denote the set of pairs g = (g_0,g_1) such that g_0 is a lower cover of g_1 and order it as follows: g <= d iff g_0 <= d_0, g_1 <= d_1, but not g_1 <= d_0. Let C(L,g) denote the connected component of g in this poset. Our main result states that C(L,g) is a semidistributive lattice if L is semidistributive, and that C(L,g) is a bounded lattice if L is bounded. Let S_n be the permutohedron on n letters and T_n be the associahedron on n+1 letters. Explicit computations show that C(S_n,a) = S_{n-1} and C(T_n,a) = T_{n-1}, up to isomorphism, whenever a is an atom. These results are consequences of new characterizations of finite join semidistributive and finite lower bounded lattices: (i) a finite lattice is join semidistributive if and only if the projection sending g in C(L) to g_0 in L creates pullbacks, (ii) a finite join semidistributive lattice is lower bounded if and only if it has a strict facet labelling. Strict facet labellings, as defined here, are generalization of the tools used by Barbut et al. to prove that lattices of Coxeter groups are bounded.