A hemi-implicative lattice is an algebra (A,∧,∨,→,1)\documentclass[12pt]{minimal}
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\begin{document}$$(A,\wedge ,\vee ,\rightarrow ,1)$$\end{document} of type (2, 2, 2, 0) such that (A,∧,∨,1)\documentclass[12pt]{minimal}
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\begin{document}$$(A,\wedge ,\vee ,1)$$\end{document} is a lattice with top and for every a,b∈A\documentclass[12pt]{minimal}
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\begin{document}$$a,b\in A$$\end{document}, a→a=1\documentclass[12pt]{minimal}
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\begin{document}$$a\rightarrow a = 1$$\end{document} and a∧(a→b)≤b\documentclass[12pt]{minimal}
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\begin{document}$$a\wedge (a\rightarrow b) \le b$$\end{document}. A new variety of hemi-implicative lattices, here named sub-Hilbert lattices, containing both the variety generated by the {∧,∨,→,1}\documentclass[12pt]{minimal}
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\begin{document}$$\{\wedge ,\vee ,\rightarrow ,1\}$$\end{document}-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\documentclass[12pt]{minimal}
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\begin{document}$$a, b \in L$$\end{document} there is an element c∈D\documentclass[12pt]{minimal}
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\begin{document}$$c \in D$$\end{document} with the property that for all d∈D\documentclass[12pt]{minimal}
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\begin{document}$$d \in D$$\end{document}, a∧d≤b\documentclass[12pt]{minimal}
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\begin{document}$$a \wedge d \le b$$\end{document} if and only if d≤c\documentclass[12pt]{minimal}
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\begin{document}$$d \le c$$\end{document} (we write a→Db\documentclass[12pt]{minimal}
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\begin{document}$$a \rightarrow _D b$$\end{document} for the element c), andS is a non void subset of L such that S is closed under →D\documentclass[12pt]{minimal}
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\begin{document}$$\rightarrow _D$$\end{document} andS, with its inherited order, is itself a lattice. Finally, the congruences of sub-Hilbert lattices are studied.
