Abstrakt: |
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: L is a bounded distributive lattice, 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 → D b for the element c), and S is a non void subset of L such that S is closed under → D and S, with its inherited order, is itself a lattice. Finally, the congruences of sub-Hilbert lattices are studied. [ABSTRACT FROM AUTHOR] |