Les treillis pseudocomple´mente´s finis

Abstract

Let L be a lattice with a least element denoted o; g(t) ∈ L is a meet pseudocomplement of t ∈ Lifx Λ t = o is equivalent to x ≤ g(t); L is meet pseudocomplemented if every element of L has a meet pseudocomplement. One defines dually the notion of join pseudocomplemented lattice. A lattice is pseudocomplemented if it is meet and join pseudocomplemented (such a lattice is sometimes called a double p -lattice, and the associated algebra a double p -algebra). These lattices have been intensively studied when they are distributive and infinite (Brouwer or Heyting lattices or algebras, Stone lattices or algebras). Examples of finite non-distributive pseudocomplemented lattices are the finite Coxeter groups endowed with the Bruhat weak ordering (especially, the ‘permutohedron’ lattice of Guilbaud and Rosenstiehl [22] is obtained with the symmetric group), and the lattice of binary bracketings (Friedman and Tamari [18]); these lattices are also complemented. We study the finite pseudocomplemented lattices, showing for instance that the pseudocomplements are obtained from the atoms and coatoms, that for complemented lattices meet pseudocomplementation is equivalent to join pseudocomplementation, and giving several results that clarify the structure of such lattices. These results can be applied especially to the permutohedron lattice and to the lattice of binary bracketings.