Sesión Lógica y ComputabilidadHemi-Nelson Algebras
Noemí Lubomirsky
Centro de Matemática La Plata (CMaLP), UNLP y CONICET, Argentina - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
This work is motivated by the common features shared by several algebraic structures associated with intuitionistic logics, including Heyting algebras, semi-Heyting algebras, subresiduated lattices, Nelson algebras, semi-Nelson algebras, and subresiduated Nelson algebras. Our aim is to provide a unified framework encompassing these varieties.
Hemi-implicative lattices [2] are defined as lattices with a greatest element, denoted by \(1\), equipped with a binary operation \(\to\) satisfying the equation \(x \to x = 1\) and the inequality \(x \wedge (x \to y) \leq y\). The latter inequality can be replaced by the following quasi-equation:
If \(z \leq x \to y\), then \(z \wedge x \leq y\).
A bounded distributive hemi-implicative lattice is a hemi-implicative lattice whose underlying lattice is distributive and has a greatest element \(1\). Many varieties of interest in algebraic logic are subvarieties of the variety of bounded distributive hemi-implicative lattices. Relevant examples include Heyting algebras, residuated lattices, RWH-algebras [3], semi-Heyting algebras, and bounded distributive Hilbert lattices (that is, Hilbert algebras [1,4] whose natural order determines a bounded distributive lattice).
Nelson’s constructive logic with strong negation, introduced and studied in [5] (see also [6,7,8]), is a non-classical logic that combines the constructive approach of positive intuitionistic logic with a classical (that is, De Morgan) negation. The algebraic semantics of this logic are given by Nelson algebras. The class of Nelson algebras, which forms a variety, has been studied since the late 1950s (initially by Rasiowa; see [6]). At the end of the 1970s, Fidel and Vakarelov independently proved that every Nelson algebra can be represented as a particular binary product (later called a twist structure) of a Heyting algebra.
The main goal of this talk is to extend this twist construction to the setting of bounded distributive hemi-implicative lattices, thereby obtaining a new variety whose members we call hemi-Nelson algebras. More precisely, an algebra \(\langle T, \wedge, \vee, \rightarrow, \sim, 0, 1 \rangle\) of type \((2,2,2,1,0,0)\) is called a hemi-Nelson algebra (or an h-Nelson algebra) if \(\langle T, \wedge, \vee, \sim, 0, 1 \rangle\) is a Kleene algebra and the following conditions hold for every \(x, y, z \in T\):
\(x \rightarrow x = 1\),
\(x \wedge (x \rightarrow y) \le x \wedge (\sim x \vee y)\),
\(\sim(x \rightarrow y) \rightarrow (x \wedge \sim y) = 1\),
\((x\wedge\sim y)\to \sim(x\to y)=1\),
\((x\wedge y \wedge (x\to y)) \to (x \wedge (x\to y)) = 1\),
\((x\wedge (x\to y)) \to (x\wedge y \wedge (x\to y)) = 1\),
\((x\to y)\vee (z\wedge\sim z)=(x\vee (z\wedge \sim z))\to (y\vee (z\wedge \sim z))\).
We show that every hemi-Nelson algebra can be represented as a twist structure over a bounded distributive hemi-implicative lattice. We also prove that, for every hemi-Nelson algebra, there exists an order isomorphism between its congruence lattice and the lattice of its h-implicative filters (which constitute a particular class of implicative filters). Furthermore, we establish an equivalence between the algebraic category of bounded distributive hemi-implicative lattices and the algebraic category of centered hemi-Nelson algebras, where a centered hemi-Nelson algebra is defined as a hemi-Nelson algebra together with a center, that is, an element fixed by the involution (such an element is necessarily unique). Finally, we apply some of the developed results to investigate several aspects of semi-Nelson algebras and subresiduated Nelson algebras.
Trabajo en conjunto con: María Paula Menchón (Universidad Nacional del Centro de la Provincia de Buenos Aires (UNICEN)) y Hernán Javier San Martín (Centro de Matemática La Plata (CMaLP), UNLP y CONICET).
Referencias
[1] Cabrer L.M., Celani S.A. and Montangie D., Representation and duality for Hilbert algebras. Central European Journal of Mathematics 7(3), 463–478 (2009).
[2] Castiglioni J.L., Fernández V., Mallea H. and San Martín H.J., Sub-Hilbert lattices. Studia Logica 111, 431-452 (2023).
[3] Celani S. and Jansana R., Bounded distributive lattices with Strict implication. Math. Log. Quart. 51, No. 3, 219--246 (2005).
[4] Diego A., Sobre álgebras de Hilbert. Notas de Lógica Matemática. Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina (1965).
[5] Nelson D., Constructible falsity. Journal of Symbolic Logic 14, 16--26 (1949).
[6] Rasiowa H., An algebraic approach to non-classical logics, volume 78 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1974).
[7] Sendlewski A., Some investigations of varieties of N-lattices. Studia Logica 43, 257--280 (1984).
[8] Vakarelov D., Notes on N-lattices and constructive logic with strong negation. Studia Logica 36, 109--125 (1977).