Abordamos diferentes problemas algebraicos en la variedad de los reticulados residuados integrales, conmutativos y acotados, así como tambi´en en la variedad de las álgebras de implicación de Lukasiewicz (subreductos implicativos de las MV-álgebras). Damos una construcción que permite sumergir todo hoop de Wajsberg en una MV- álgebra y la utilizamos para desarrollar dualidades topológicas para ciertas clases de hoops de Wajsberg y para caracterizar los hoops de Wajsberg k-valuados libres. Estudiamos la descomponibilidad de las álgebras libres para diferentes subvariedades de reticulados residuados, probando la indescomponibilidad en ciertos casos y caracterizando las subvariedades de reticulados residuados pseudocomplementados que poseen sus álgebras libres descomponibles. Estudiamos tambi´en los elementos regulares de un reticulado residuado, introduciendo la noción de variedad regular y estableciendo sus conexiones con la traducción negativa de Kolmogorov. Obtenemos una representaci´on sencilla de las álgebras de implicación de Lukasiewicz finitas como crecientes en productos de MV-cadenas finitas y damos una dualidad topol ógica intr´ınseca para las álgebras de implicación. Caracterizamos la permutabilidad de congruencias en dichas álgebras, probamos que todas las subcuasivariedades son variedades y mostramos que todos los miembros finitos de esta variedad son débilmente proyectivos. Estudiamos tambi´en las clases algebraicamente expandibles en esta variedad, así como también las funciones algebraicas, especialmente para la subvariedad generada por la cadena de tres elementos. / We deal with different algebraic problems in the variety of integral, commutative, bounded residuated lattices, as well as in the variety of Lukasiewicz implication algebras (implicative subreducts of MV-algebras). We give a construction that allows us to embed anyWajsberg hoop into an MV-algebra and use it to develop topological dualities for certain classes of Wajsberg hoops and to characterize the free k-valued Wajsberg hoops. We study the decomposability of free algebras in different subvarieties of residuated lattices, establishing the indecomposability for some cases and characterizing the subvarieties of pseudocomplemented residuated lattices whose free algebras are decomposable. We also study the regular elements of a residuated lattices, introducing the notion of regular variety and establishing connections with the negative Kolmogorov translation. We obtain a simple representation of finite Lukasiewicz implication algebras as upsets in products of finite MV-chains and give an intrinsic topological duality for implication algebras. We characterize congruence permutability for these algebras, prove that any subquasivariety is a variety and show that every finite member of this variety is weakly projective. We also study algebraically expandable classes in this variety, as well as algebraic functions, especially for the subvariety generated by the three-element chain.

Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2018-03-02T23:39:14Z No. of bitstreams: 1 ThiagoNascimentoDaSilva_DISSERT.pdf: 675458 bytes, checksum: 9123812e69a846020d3cd6346e530e1e (MD5) / Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2018-03-13T18:55:45Z (GMT) No. of bitstreams: 1 ThiagoNascimentoDaSilva_DISSERT.pdf: 675458 bytes, checksum: 9123812e69a846020d3cd6346e530e1e (MD5) / Made available in DSpace on 2018-03-13T18:55:45Z (GMT). No. of bitstreams: 1 ThiagoNascimentoDaSilva_DISSERT.pdf: 675458 bytes, checksum: 9123812e69a846020d3cd6346e530e1e (MD5) Previous issue date: 2018-01-25 / Al?m da mais conhecida l?gica de Nelson (?3) e da l?gica paraconsistente de Nelson (?4), David Nelson introduziu no artigo de 1959 "Negation and separation of concepts in constructive systems", com motiva??es de aritm?tica e construtividade, a l?gica que ele chamou de "?". Naquele trabalho, a l?gica ? definida por meio de um c?lculo (que carece crucialmente da regra de contra??o) tendo infinitos esquemas de regras, e nenhuma sem?ntica ? fornecida. Neste trabalho n?s tomamos o fragmento proposicional de ?, mostrando que ele ? algebriz?vel (de fato, implicativo) no sentido de Blok & Pigozzi com respeito a uma classe de reticulados residuados involutivos. Assim, fornecemos a primeira sem?ntica para ? (que chamamos de ?-?lgebras), bem como um c?lculo estilo Hilbert finito equivalente ? apresenta??o de Nelson. Fornecemos um algoritmo para construir ?-?lgebras a partir de ?-?lgebras ou reticulados implicativos e demonstramos alguns resultados sobre a classe de ?lgebras que introduzimos. N?s tamb?m comparamos ? com outras l?gicas da fam?lia de Nelson, a saber, ?3 e ?4. / Besides the better-known Nelson logic (?3) and paraconsistent Nelson logic (?4), in Negation and separation of concepts in constructive systems (1959) David Nelson introduced a logic that he called ?, with motivations of arithmetic and constructibility. The logic was defined by means of a calculus (crucially lacking the contraction rule) having infinitely many rule schemata, and no semantics was provided for it. We look in the present dissertation at the propositional fragment of ?, showing that it is algebraizable (in fact, implicative) in the sense of Blok and Pigozzi with respect to a class of involutive residuated lattices. We thus provide the first known algebraic semantics for ?(we call them of ?-algebras) as well as a finite Hilbert-style calculus equivalent to Nelson?s presentation. We provide an algorithm to make ?-algebras from ?-algebras or implicative lattices and we prove some results about the class of algebras which we have introduced. We also compare ? with other logics of the Nelson family, that is, ?3 and ?4.