Compatibly involutive residuated lattices and the Nelson identity
Nelson’s constructive logic with strong negation N3 can be presented (to within definitional equivalence) as the axiomatic extension NInFL ew of the involutive full Lambek calculus with exchange and weakening by the Nelson axiom[Figure not available: see fulltext.] The algebraic counterpart of NInFL...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Institución: | Universidad Nacional de Educación a Distancia |
| Repositorio: | e-spacio. Repositorio Institucional de la UNED |
| Idioma: | inglés |
| OAI Identifier: | oai:e-spacio.uned.es:20.500.14468/24645 |
| Acceso en línea: | https://hdl.handle.net/20.500.14468/24645 |
| Access Level: | acceso abierto |
| Palabra clave: | 11 Lógica Nelson algebra Nelson logic compatibly involutive residuated lattices congruence orderable Fregean |
| Sumario: | Nelson’s constructive logic with strong negation N3 can be presented (to within definitional equivalence) as the axiomatic extension NInFL ew of the involutive full Lambek calculus with exchange and weakening by the Nelson axiom[Figure not available: see fulltext.] The algebraic counterpart of NInFL ew is the recently introduced class of Nelson residuated lattices. These are commutative integral bounded residuated lattices ⟨ A; ∧ , ∨ , ∗ , ⇒ , 0 , 1 ⟩ that: (i) are compatibly involutive in the sense that ∼ ∼ a= a for all a∈ A, where ∼ a: = a⇒ 0 , and (ii) satisfy the Nelson identity, namely the algebraic analogue of (Nelson ⊢ ), viz.(x⇒(x⇒y))∧(∼y⇒(∼y⇒∼x))≈x⇒y.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 A is a Nelson residuated lattice iff for all a, b∈ A, the congruence condition ΘA(0,a)=ΘA(0,b)andΘA(1,a)=ΘA(1,b)impliesa=bholds. 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. |
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