Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 61-63). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

Algebraic theories describe mathematical structures that are defined in terms of operations and equations, and are extremely important throughout mathematics. Many generalisations of the classical notion of an algebraic theory have sprung up for use in different mathematical contexts; some examples include Lawvere theories, monads, PROPs and operads. The first central notion of this thesis is a common generalisation of these, which we call a proto-theory. The purpose of an algebraic theory is to describe its models, which are structures in which each of the abstract operations of the theory is given a concrete interpretation such that the equations of the theory hold. The process of going from a theory to its models is called semantics, and is encapsulated in a semantics functor. In order to define a model of a theory in a given category, it is necessary to have some structure that relates the arities of the operations in the theory with the objects of the category. This leads to the second central notion of this thesis, that of an interpretation of arities, or aritation for short. We show that any aritation gives rise to a semantics functor from the appropriate category of proto-theories, and that this functor has a left adjoint called the structure functor, giving rise to a structure{semantics adjunction. Furthermore, we show that the usual semantics for many existing notions of algebraic theory arises in this way by choosing an appropriate aritation. Another aim of this thesis is to find a convenient category of monads in the following sense. Every right adjoint into a category gives rise to a monad on that category, and in fact some functors that are not right adjoints do too, namely their codensity monads. This is the structure part of the structure{semantics adjunction for monads. However, the fact that not every functor has a codensity monad means that the structure functor is not defined on the category of all functors into the base category, but only on a full subcategory of it. This deficiency is solved when passing to general proto-theories with a canonical choice of aritation whose structure{semantics adjunction restricts to the usual one for monads. However, this comes at a cost: the semantics functor for general proto-theories is not full and faithful, unlike the one for monads. The condition that a semantics functor be full and faithful can be thought of as a kind of completeness theorem | it says that no information is lost when passing from a theory to its models. It is therefore desirable to retain this property of the semantics of monads if possible. The goal then, is to find a notion of algebraic theory that generalises monads for which the semantics functor is full and faithful with a left adjoint; equivalently the semantics functor should exhibit the category of theories as a re ective subcategory of the category of all functors into the base category. We achieve this (for well-behaved base categories) with a special kind of proto-theory enriched in topological spaces, which we call a complete topological proto-theory. We also pursue an analogy between the theory of proto-theories and that of groups. Under this analogy, monads correspond to finite groups, and complete topological proto-theories correspond to profinite groups. We give several characterisations of complete topological proto-theories in terms of monads, mirroring characterisations of profinite groups in terms of finite groups.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O propÃsito desta dissertaÃÃo à apresentar aplicaÃÃes da Teoria dos Grafos à Teoria dos Grupos. De posse do grafo associado a um grupo finito, nÃs obtemos vÃrios resultados interessantes sobre a estrutura do grupo analisando tal grafo à luz de tÃcnicas-padrÃo da Teoria dos Grafos. Mais precisamente, os nÃmeros cromÃtico e de independÃncia do grafo de um grupo finito nos permitem estimar a cardinalidade mÃxima de um subgrupo abeliano do mesmo, bem como o tamanho mÃnimo possÃvel de um subconjunto do grupo formado por elementos que nÃo comutam dois a dois; no caso de grupos finitos abelianos, nÃs tambÃm estudamos seus subconjuntos livres de somas. / This report deals with applications of Graph Theory to Group Theory. Once we construct the graph associated to a finite group, we get several interesting results on the group structure by analysing its associated graph with the help of various standard graph-theoretic tools. More precisely, the chromatic and independence numbers of the graph of a finite group allows us to estimate the maximal cardinality of an abelian subgroup of it, as well as the minimal size of a subset of the group, all of whose elements donât commute in pairs; for finite abelian groups, we also study their free-sum subsets.

The idea of keychains, an (n+1)-tuple of non-increasing real numbers in the unit interval always including 1, naturally arises in study of finite fuzzy set theory. They are a useful concept in modeling ideas of uncertainty especially those that arise in Economics, Social Sciences, Statistics and other subjects. In this thesis we define and study some basic properties of keychains with reference to Partially Ordered Sets, Lattices, Chains and Finite Fuzzy Sets. We then examine the role of keychains and their lattice diagrams in representing uncertainties that arise in such problems as in preferential voting patterns, outcomes of competitions and in Economics - Preference Relations.

We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.

In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .

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