Sub-Hilbert Lattices

Abstract

A hemi-implicative lattice is an algebra (A, ∧ , ∨ , → , 1) of type (2, 2, 2, 0) such that (A, ∧ , ∨ , 1) is a lattice with top and for every a, b∈ A, a→ a= 1 and a∧ (a→ b) ≤ b. A new variety of hemi-implicative lattices, here named sub-Hilbert lattices, containing both the variety generated by the { ∧ , ∨ , → , 1 } -reducts of subresiduated lattices and that of Hilbert lattices as proper subvarieties is defined. It is shown that any sub-Hilbert lattice is determined (up to isomorphism) by a triple (L, D, S) which satisfies the following conditions: 1.L is a bounded distributive lattice,2.D is a sublattice of L containing 0, 1 such that for each a, b∈ L there is an element c∈ D with the property that for all d∈ D, a∧ d≤ b if and only if d≤ c (we write a→ Db for the element c), and3.S is a non void subset of L such that i.S is closed under → D andii.S, with its inherited order, is itself a lattice. Finally, the congruences of sub-Hilbert lattices are studied.