De Morgan monoids are algebraic structures that model certain non-classical logics. The variety DMM of all De Morgan monoids models the relevance logic Rt (so-named because it blocks the derivation of true conclusions from irrelevant premises). The so-called subvarieties and subquasivarieties of DMM model the strengthenings of Rt by new logical axioms, or new inference rules, respectively. Meta-logical problems concerning these stronger systems amount to structural problems about (classes of) De Morgan monoids, and the methods of universal algebra can be exploited to solve them. Until now, this strategy was under-developed in the case of Rt and DMM. The thesis contributes in several ways to the filling of this gap. First, a new structure theorem for irreducible De Morgan monoids is proved; it leads to representation theorems for the algebras in several interesting subvarieties of DMM. These in turn help us to analyse the lower part of the lattice of all subvarieties of DMM. This lattice has four atoms, i.e., DMM has just four minimal subvarieties. We describe in detail the second layer of this lattice, i.e., the covers of the four atoms. Within certain subvarieties of DMM, our description amounts to an explicit list of all the covers. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids. Thereafter, we use these insights to identify strengthenings of Rt with certain desirable meta-logical features. In each case, we work with the algebraic counterpart of a meta-logical property. For example, we identify precisely the varieties of De Morgan monoids having the joint embedding property (any two nontrivial members both embed into some third member), and we establish convenient sufficient conditions for epimorphisms to be surjective in a subvariety of DMM. The joint embedding property means that the corresponding logic is determined by a single set of truth tables. Epimorphisms are related to 'implicit definitions'. (For instance, in a ring, the multiplicative inverse of an element is implicitly defined, because it is either uniquely determined or non-existent.) The logical meaning of epimorphism-surjectivity is, roughly speaking, that suitable implicit definitions can be made explicit in the corresponding logical syntax. / Thesis (PhD)--University of Pretoria, 2020. / DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) / Mathematics and Applied Mathematics / PhD / Unrestricted

Phosphate deposits in the western United States have been the subject of considerable discussion in the last decade, and during the last few years much interest has been aroused in them. Phosphate was first discovered in this area in 1889, but it was not until 1906 that the United States Government took active interest in the study of these deposits. The investigation by the government of the occurrence of western phosphate was begun by Weeks and Ferrier (10), and later carried on by others. On December 9, 1908 the western phosphate reserve was created and many acres of land were withdrawn from all kinds of entry. In the summer of 1909 two parties were detailed by the United States Geological Survey to examine lands withdrawn from public entry.

Our civilization depends in great measure on the reading process and there is a need for attaining greater skill in reading. Being able to read well has become one criteria for measuring the extent of a person's education. Every year seems to increase the reading demands made upon students as well as adults. Reading was one of the three R's that made up the curriculum of the early schools in our country.

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El objetivo de esta tesis es abordar distintos problemas algebraicos acerca de algunas subvariedades de las álgebras de De Morgan Heyting y de las álgebras pseudocomplementadas de Kleene utilizando dualidades topológicas tipo Priestley correspondientes a dichas variedades. Se investiga la sucesión de subvariedades SDHn de las álgebras de De Morgan Heyting caracterizadas por la identidad xn(1*) = x(n+1)(1*) definidas por H.P. Sankappanavar en [26]. Se obtienen condiciones necesarias y sufi- cientes sobre el espacio de filtros primos para que un álgebra de De Morgan Heyting pertenezca a la variedad SDH1 y se caracterizan las álgebras subdirectamente irreducibles y simples de dicha variedad. Todos estos resultados son extendidos para las álgebras finitas en el caso general SDHn. La clase de las álgebras de Boole es un ejemplo familiar de álgebras de Heyting y es bien conocido que existe una correspondencia entre las subálgebras de un álgebra de Boole y ciertas relaciones de equivalencia definidas sobre su espacio Booleano (ver, por ejemplo [13]). En esta tesis se extiende esta correspondencia tanto para la clase de las álgebras de Heyting como para la clase de las álgebras de De Morgan Heyting, es decir, se caracterizan las subálgebras de las álgebras de Heyting y de De Morgan Heyting definiendo ciertas relaciones de equivalencia sobre los espacios topológicos de sus respectivas representaciones tipo Priestley. Como caso particular de este resultado, se obtiene la caracterización para subálgebras maximales de las álgebras de Heyting finitas dada por M. Adams en [2]. Se estudian las álgebras subdirectamente irreducibles en la variedad PCDM de las álgebras pseudocomplementadas de De Morgan a través de sus pm-espacios. Se introduce la noción de body de un álgebra L 2 PCDMy se caracteriza completamente Body(L) cuando L es subdirectamente irreducible, directamente indescomponible o simple. Como consecuencia de esto, en el caso particular de las álgebras pseudocomplementadas de Kleene, surgen naturalmente tres subvariedades de la misma para las cuales se determinan identidades que las caracterizan. Se define la subvariedad BPK, de particular interés ya que sus álgebras subdirectamente irreducibles son suma ordinal de álgebras de Boole y cadenas, realizándose un estudio de la misma. Se determina completamente el reticulado de sus subvariedades y se encuentran bases ecuacionales para cada una de ellas. Una de estas subvariedades, llamada BPK0 es aquella cuyos miembros subdirectamente irreducibles son de la forma B B, donde B es un álgebra de Boole. La última parte de la tesis está destinada al estudio de la variedad BPK0 resolviéndose problemas tales como la obtención de las álgebras libres con una cantidad finita de generadores libres y la descripción completa del reticulado de cuasivariedades junto con una base de cuasi-identidades para cada cuasivariedad. / The objective of this thesis is to study several algebraic problems regarding some subvarieties of De Morgan Heyting algebras and pseudocomplemented Kleene algebras using the corresponding Priestley dualities as a main tool. We focus on the sequence of subvarieties SDHn, which consist of the De Morgan Heyting algebras characterized by the identity xn(1*) =x(n+1)(1*), as defined by H. P. Sankappanavar in [26]. We give necessary and suficient conditions on the space of prime filters for a De Morgan Heyting algebra to belong to the variety SDH1. We also characterize the subdirectly irreducible and simple members of this variety. These results are all further extended for finite algebras in the general case of the varieties SDHn. The class of Boolean algebras is a familiar example of Heyting algebras and it is well known that there exists a correspondence between subalgebras of a Boolean algebra and certain equivalence relations on its Boolean space (see, for example, [13]). In this thesis, we extend this correspondence both for the class of Heyting algebras and for the class of De Morgan Heyting algebras, that is, we characterize the subalgebras of a Heyting algebra and a De Morgan Heyting algebra by defining certain equivalence relations on their respective Priestley spaces. The characterization of maximal subalgebras in finite Heyting algebras given by M. Adams in [2] follows now as a special case of our characterization. We also study the subdirectly irreducible members of the variety PCDM of pseudocomplemented De Morgan algebras in terms of their pm-spaces. We introduce the notion of body of an algebra L 2 PCDM and characterize completely the body of L when L is subdirectly irreducible, directly indecomposable or simple. As a consequence of this, in the case of pseudocomplemented Kleene algebras, three special subvarieties arise naturally, for which we give explicit identities that characterize them. We also define the variety BPK which is of particular interest because its subdirectly irreducible algebras are ordinal sums of Boolean algebras and chains. We study this variety in depth. We determine the whole subvariety lattice and find explicit equational bases for each of the subvarieties. The subdirectly irreducible members of one of these subvarieties, called BPK0, are of the form B B, where B is a Boolean algebra. The last part of this thesis is devoted to the study of this variety: we characterize the finitely generated free algebras and give a full description of the quasivariety lattice as well as the corresponding quasi-equational basis for each of the quasivarieties.

The late Victorian artist Evelyn De Morgan's paintings have been analyzed and interpreted through the lens of her many stylistic influences by past critics and current art historians. This thesis seeks to restore 19th-century Spiritualism as the central influence on the subject matter and style of De Morgan's paintings. This is particularly true of works concerned with the struggles of mortal life and the moment of death, based on her anonymously published text The Result of an Experiment. Victorian mourning rituals, Spiritualism, and the writings of Swedenborg served to draw out the specific Spiritualist symbols within De Morgan's paintings. A detailed analysis of six paintings concerned with the path of mortal life and death revealed De Morgan's Spiritualist beliefs about a hopeful death after her experiment with spirit communication.

The thesis introduces a logic which combines intuitionistic implication with de Morgan negation in a way which conservatively extends both classical and intuitionistic logic. This logic is the intuitionistic counterpart of the four-valued Belnap-Dunn logic. In relation to this logic, we study de Morgan algebras and their expansions, in particular their expansion with a constant representing inconsistency. We prove a duality for such algebras extending the Priestley duality. We also introduce a weak notion of modal algebra and prove a duality for such algebras. We then define analytic sequent calculi for various logics of de Morgan negation. Powered by TCPDF (www.tcpdf.org)

In this thesis we present the Stone representation theorem, generally known as Stone duality in the point-free context. The proof is choice-free and, since we do not have to be concerned with points, it is by far simpler than the original. For each infinite cardinal κ we show that the counter- part of the κ-complete Boolean algebras is constituted by the κ-basically disconnected Stone frames. We also present a precise characterization of the morphisms which correspond to the κ-complete Boolean homomorphisms. Although Booleanization is not functorial in general, in the part of the dual- ity for extremally disconnected Stone frames it is, and constitutes an equiv- alence of categories. We finish the thesis by focusing on the De Morgan (or extremally disconnected) frames and present a new characterization of these by their superdense sublocales. We also show that in contrast with this phenomenon, a metrizable frame has no non-trivial superdense sublocale; in other words, a non-trivial Čech-Stone compactification of a metrizable frame is never metrizable. 1

University of Macau / Faculty of Arts and Humanities. / Department of English