Compatibly involutive residuated lattices and the Nelson identity

Abstract

Nelson’s constructive logic with strong negation $$\mathbf {N3}$$ can be presented (to within definitional equivalence) as the axiomatic extension $$\mathbf {NInFL}_{ew}$$ of the involutive full Lambek calculus with exchange and weakening by the Nelson axiom The algebraic counterpart of $$\mathbf {NInFL}_{ew}$$ is the recently introduced class of Nelson residuated lattices. These are commutative integral bounded residuated lattices $$\langle A; \wedge , \vee , *, \Rightarrow , 0, 1 \rangle $$ that: (i) are compatibly involutive in the sense that $$\mathop {\sim }\mathop {\sim }a = a$$ for all $$a \in A$$ , where $$\mathop {\sim }a := a \Rightarrow 0$$ , and (ii) satisfy the Nelson identity, namely the algebraic analogue of (Nelson $$_{\vdash }$$ ), viz. Nelson $$\begin{aligned} \bigl (x \Rightarrow (x \Rightarrow y)\bigr ) \wedge \bigl (\mathop {\sim }y \Rightarrow (\mathop {\sim }y \Rightarrow \mathop {\sim }x)\bigr ) \approx x \Rightarrow y. \end{aligned}$$ The present paper focuses on the role played by the Nelson identity in the context of compatibly involutive commutative integral bounded residuated lattices. We present several characterisations of the identity (Nelson) in this setting, which variously permit us to comprehend its model-theoretic content from order-theoretic, syntactic, and congruence-theoretic perspectives. Notably, we show that a compatibly involutive commutative integral bounded residuated lattice $$\mathbf {A}$$ is a Nelson residuated lattice iff for all $$a, b \in A$$ , the congruence condition $$\begin{aligned} \varTheta ^{\mathbf {A}}(0, a) = \varTheta ^{\mathbf {A}}(0, b) \quad \text {and} \quad \varTheta ^{\mathbf {A}}(1, a) = \varTheta ^{\mathbf {A}}(1, b) \quad \text {implies} \quad a = b \end{aligned}$$ holds. This observation, together with others of the main results, opens the door to studying the characteristic property of Nelson residuated lattices (and hence Nelson’s constructive logic with strong negation) from a purely abstract perspective.