Sub-Hilbert Lattices

Abstract

A hemi-implicative lattice is an algebra $(A,\wedge ,\vee ,\rightarrow ,1)$ of type (2, 2, 2, 0) such that $(A,\wedge ,\vee ,1)$ is a lattice with top and for every $a,b\in A$, $a\rightarrow a = 1$ and $a\wedge (a\rightarrow b) \le b$. A new variety of hemi-implicative lattices, here named sub-Hilbert lattices, containing both the variety generated by the $\{\wedge ,\vee ,\rightarrow ,1\}$-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 $a, b \in L$ there is an element $c \in D$ with the property that for all $d \in D$, $a \wedge d \le b$ if and only if $d \le c$ (we write $a \rightarrow _D b$ for the element c), and

• 3. S is a non void subset of L such that

  • i. S is closed under $\rightarrow _D$ and

  • ii. S, with its inherited order, is itself a lattice.

Finally, the congruences of sub-Hilbert lattices are studied.