Álgebras de De Morgan pseudocomplementadas modales 4-valuadas

Las álgebras de De Morgan pseudocomplementadas fueron consideradas por primera vez por A. Romanowska ([66]) quien las denominó pM−álgebras y caracterizó las álgebras subdirectamente irreducibles finitas. Posteriormente, H. Sankappanavar ([67, 68]) continuó con el estudio de las pM−álgebras examinando las congruencias y caracterizando todas las subdirectamente irreducibles. Por otra parte, A. V. Figallo y P. Landini ([23, 21]) con el propósito de presentar distintas axiomáticas para las álgebra tetravalente modales ([42, 43]), mostraron que las pM−álgebras que verifican la condición adicional x V~x

De Morgan pseudocomplemented algebras were first considered by A. Romanowska ([66]) who called them pM−algebras and characterized the finite subdirectly irreducible algebras. Later on, H. Sankappanavar ([67, 68]) continued studying pM−algebras by examining congruences and characterizing all the subdirectly irreducible algebras. On the other hand, A. V. Figallo and P. Landini ([23, 21]), with the aim of presenting different axiomatic for tetravalent modal algebras ([42, 43]), they proved that pM−algebras verifying the additional condition x_V~ x < _ xVx* admit a tetravalent modal algebra structure. Hence, they called them De Morgan pseudocomplemented modal algebras, or mpM−algebras, for short. Our aim in this thesis is to study in deep the variety mpM of mpM−algebras. More precisely, we have organized this work in four chapters. In Chapter I, basic definitions are provided and we do also a review of the most important results in universal algebra. Furthermore, we have also included a brief discussion on Priestley’s dualities for bounded distributive lattices and p−algebras ([60, 61, 63]). Finally, we describe W. Cornish and P. Fowler’s duality ([18, 19]) for De Morgan algebras. These topics have been included not only to simplify the reading but also to fix the notations and the definitions that we will use in this volume. In Chapter II, we began the study of mpM−algebras. Here, we boarded the problem of characterizing the subdirectly irreducible members of this variety. To this aim, we determine a topological duality for these algebras which allowed us to characterize the congruence lattice.We must point out that this duality is strongly used throughout all this work. Furthermore, we prove that mpM−algebras constitute a locally finite, semisimple, residually small and residually finite variety. In the last section of this chapter we obtain, by means of algebraic techniques, other characterizations of the congruences by means of special subsets of the algebra. Some of the above results were presented in the Annual Meeting of the Uni´on Matemática Argentina in 2004 and 2006. In Chapter III, and in order to obtain more information on the variety mpM, we carried out a detailed study of the principal congruences. First, we indicate two descriptions of them by means certain subsets of the associated space to an mpM−algebra, which allowed us to conclude that they constitute a Boolean algebra. Next we show, among other results, that mpM is a discriminator variety which also provided us many properties of mpM−congruences. Later on, we prove that principal and Boolean congruences coincide and this statement allows us to determine the number of congruences in the finite mpM−algebras. By the end of this chapter, we determine the ternary discriminator polynomial for this variety and we also establish an equational description of the principal congruences. It is worth mentioning that some of the topics presented in this chapter were previously discussed at the XIII Latin American Symposium on Mathematical Logic, Oaxaca, Mexico, and in the Annual Meeting of the Unión Matem´atica Argentina in 2006 and 2007 respectively. Chapter IV consists of 2 sections. In the first one, we focus our study on the properties of finite and finitely generated mpM−algebras. In the second one, we determine the structure of the free mpM−algebras with a finite set of free generators. Finally, we indicate a formula which allows us to calculate the cardinal number of the free mpM−algebras in terms of the number of the free generators of the algebras. Some of the results of this chapter were presented at the Annual Meeting of the Uni´on Matem´atica Argentina in 2008, Some of the topics of this thesis have been accepted for publication in ([24]).