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: _L_ is a bounded distributive lattice, _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 _S_ is a non void subset of _L_ such that _S_ is closed under $\rightarrow _D$ and _S_, with its inherited order, is itself a lattice. Finally, the congruences of sub-Hilbert lattices are studied.