A Categorical Equivalence Motivated by Kalman's Construction

SAGASTUME MARTA; SAN MARTÍN HERNÁN JAVIER

STUDIA LOGICA

Springer

Año: 2016 vol. 104 p. 185 - 206

0039-3215

In [S-SM] we have studied the relationship between the infinite-valued logic L of Lukasiewicz and a new system that we call L^{bullet}. The models of the logic L^{bullet} are the objects of the algebraic category MV^{bullet}, which is dually equivalent to the category MV of MV-algebras by means of the functor K^{bullet} whose adjoint is a functor Kappa (see [camesa2,CLS]). Also, we define a more general deductive system whose equivalent algebraic semantics is a variety of involutive residuated lattices in which there is a unary map kappa that plays an important role.In particular, in [S-SM] we found a link between the Lindenbaum algebra of L, A_{{MV}}, and that of L^{bullet}, A_{{MV}^{bullet}}, by means of an isomorphism kappa(A_{{MV^{bullet}}) =  A_{{MV}}/I, for some ideal I of A_{{MV}}. This fact and related work by L. Monteiro and I. Viglizzo [V1, V2] motivate us to define an extension of the functor K^{bullet}. The objects of the domain category of the new functor are pairs (A,I), where A is an integral commutative residuated lattice which has an involution and I is an ideal of A.References[camesa2] J.L. Castiglioni, M. Menni and M. Sagastume, On some categories of involutive centered residuated lattices.Studia Logica 90, no. 1, 93--124 (2008). [CLS] J.L. Castiglioni, R. Lewin and M. Sagastume, On a definition of a variety of monadic l-groups. Studia Logica 102, no. 1, 67--92 (2014).[V1]  L. Monteiro and I. Viglizzo, Construction of Nelson algebras.  Personal comunication (1997).[S-SM]  M. ~Sagastume and H.J. San Martín, The logic L{bullet}. Accepted in Math. Logic Quarterly.[V2] I. Viglizzo, Algebras de Nelson. Tesis de Magíster, UNS, Bahía Blanca, Buenos Aires, Argentina (1999).