Cayley lattices of finite Coxeter groups are bounded

In 1994, Le Conte de Poly-Barbut has proved that Cayley lattices associated with finite Coxeter groups (for short Coxeter lattices) are semidistributive, which is a strong algebraic property. The semidistributivity of a lattice implies that there exists a particular bijection — called the double-arrow relation—between the join-irreducible and the meet-irreducible elements of the lattice. erval doubling is a simple constructive operation which applies on a lattice L and an interval of L, and which constructs a new “bigger” lattice. in result of this paper says that all Coxeter lattices are bounded, a property characterized by the fact that they are obtainable by a finite series of interval doublings, starting with the one-element lattice. This property implies the semidistributivity of Coxeter lattices and brings up to light some strong and rich relations between the reflections of the Coxeter group, the interval doubling construction and the double-arrow relation.