On Weak Lewis Distributive Lattices

In this paper we study the variety \(\textsf{WL}\) $\mathsf{WL}$ of bounded distributive lattices endowed with an implication, called weak Lewis distributive lattices. This variety corresponds to the algebraic semantics of the \(\{\vee ,\wedge ,\Rightarrow ,\bot ,\top \}\) $\{\vee ,\wedge ,\Rightarrow ,\perp ,\top \}$ -fragment of the arithmetical base preservativity logic \(\mathsf {iP^{-}}\) ${\mathsf{iP}}^{-}$ . The variety \(\textsf{WL}\) $\mathsf{WL}$ properly contains the variety of bounded distributive lattices with strict implication, also known as weak Heyting algebras. We introduce the notion of WL-frame and we prove a representation theorem for WL-lattices by means of WL-frames. We extended this representation to a topological duality by means of Priestley spaces endowed with a special neighbourhood relation between points and closed upsets of the space. These results are applied in order to give a representation and a topological duality for the variety of weak Heyting–Lewis algebras, i.e., for the algebraic semantics of the arithmetical base preservativity logic \(\textsf{iP}^{-}\) ${\mathsf{iP}}^{-}$ .