The lattice of strict completions of a finite poset

Abstract

For a given finite poset (P, ≤), we construct strict completions of P which are models of all finite lattices L such that the set of join-irreducible elements of L is isomorphic to P . This family of lattices, M P , turns out to be itself a lattice, which is lower bounded and lower semimodular. We determine the join-irreducible elements of this lattice. We relate properties of the lattice M P to properties of our given poset P , and in particular we characterize the posets P for which |M P | ≤ 2. Finally we study the case where M P is distributive.

Going down in (semi)lattices of finite moore families and convex geometries

“…Observe that, for any given poset (P, ), there exists at least one Moore family G such that J(G ) is isomorphic to (P, ), namely the set O(P ) of all ideals of (P, ), but there exist in general several such Moore families (the so-called strict completions of P , see Bordalo and Monjardet [4], 2002).…”

“…The poset M P is a lattice which was studied in Bordalo and Monjardet [4], 2002. Its maximum is the lattice O(P ) of all ideals of P .…”

“…In fact, the following definitions and results allow us to make more precise the covering relation of M P and G (the notions of forced and M P -deletable ideals are in Bordalo and Monjardet [4], 2002). In these definitions and results, G ′ (rather than G ) will denote an arbitrary Moore family belonging to one of our four posets of Moore families, and G will always denote a Moore family covered by G ′ in this poset.…”

“…The set M of all these families defined on a set P and ordered by set inclusion is a lattice studied by many authors (see Caspard and Monjardet [7], 2003). The set M P of all Moore families whose poset of join-irreducible elements is isomorphic to a poset (P, ) (where is any given partial order defined on P ) is a lattice studied in Bordalo and Monjardet [4], 2002. Convex geometries are a significant class of Moore families appearing in many domains.…”

An algebra for specifying valid compound terms in faceted taxonomies

“…The broader and the narrower compound terms of s are defined as follows: Br(s) = {s ∈ P (T ) | s s } and Nr(s) = {s ∈ P (T ) | s s}. In lattice theory terms [6], Br(s) is the principal order filter generated by s and Nr(s) is the principal order ideal generated by s.…”