Hypermatrices are defined. Elementary operations and properties are defined and discussed. A 4-ary Multiplication is defined for hypermatrices, consisting of multilinear mappings from ordered 4-tuples of hypermatrices to hypermatrices. This multiplication is the only such mapping satisfying two basic properties which we should like such an operation to have. Various properties and characterizations of Multiplication are discussed. Equivalence Classes of hypermatrices are defined and discussed. Starting with equivalence classes of a general nature, we are led to the definition of various types of Hyperdeterminants, themselves considered as being equivalence classes of hypermatrices. Operators and operations on hypermatrices are extended to hyperdeterminants. A generalization of the Cauchy-Binet Theorem for matrices is seen to hold for hypermatrices and their associated hyperdeterminants. Special systems of hypermatrices are seen to constitute generalizations of the Complex and Quaternion Algebras, and some properties of these are discussed. / Science, Faculty of / Mathematics, Department of / Graduate

by Jin Mai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 74-79). / Acknowledgement --- p.i / Abstract --- p.ii / Chapter 1. --- Pseudovarieties of finite algebras --- p.1 / Chapter 2. --- Algebraic automata and formal languages theory --- p.19 / Chapter 3. --- M-varieties and S-varieties --- p.36 / Chapter 4. --- The dot-depth hierarchy --- p.48 / Chapter 5. --- Operators P and P' --- p.62 / References --- p.74

Portion of Thesis (Ph. D.)--University of Michigan, 1941. / Cover title. "Reprinted from Annals of Mathematics, vol. 43, no. 1, January, 1942." Includes bibliographical references.

Ph.D. (Mathematics) / This thesis reports on an interdisciplinary research programme: an investigation of power structures, and their applications in various fields. A power construction is an attempt to lift whatever structure may exist between the elements of a set to subsets of that set. The notions of structure considered here are algebraic, relational and topological. It is shown how power constructions are useful in a number of contexts in Mathematics, in Logic, in Computer Science and in the Philosophy of Science. The thesis is therefore an exercise in what may be called lateral research, where the aim is to look horizontally across disciplinary boundaries, identify common basic concepts, and use these to fertilise each field with results from the others. This differs from the more common vertical research method, the two manifestations of which (in Mathematics, in particular) are specialisation and generalisation. To specialise means to narrow down the field of investigation, as with a group theorist studying specifically Abelian groups. Generalisation moves in the opposite direction - one may attempt, for example, to generalise a result first proved for Abelian groups to the case of arbitrary groups. But, whether narrowing down or opening up, in vertical research mode it is the concept alone which is under investigation - its own nature, rather than its relationships to other concepts. In particular, vertical research pays little attention to the occurrence and application of the concept under investigation in other disciplines. It is sad that 'research', in Mathematics, is often unthinkingly equated with 'vertical research'. This is detrimental to scholarship in at least two ways. One is the training of new scientists - more particularly, of new PhD's. It is ironic that though the requirement for a PhD is almost universally held to be 'original research', or 'a contribution to knowledge', few things are in fact more orthodox and conformist than a PhD thesis. Here I refer not just to presentation (uniformity of which may be beneficial), but to methodology: few PhD candidates would dare to prejudice their chances with unpredictable examiners by venturing outside the paradigm of vertical research. A second effect (which is also a cause) of equating 'research' with 'vertical research' is the allocation of research funding. Project proposals and grant applications must be evaluated; this is usually done by peer review, and it seems clear that referees' reports emanating from a smallish fraternity of specialists will have a more enthusiastic ring than...

This thesis attempts a synthesis of two important and fast developing branches of mathematics, namely universal algebra and fuzzy set theory. Given an abstract algebra [X,F] where X is a non-empty set and F is a set of finitary operations on X, a fuzzy algebra [I×,F] is constructed by extending operations on X to that on I×, the set of fuzzy subsets of X (I denotes the unit interval), using Zadeh's extension principle. Homomorphisms between fuzzy algebras are defined and discussed. Fuzzy subalgebras of an algebra are defined to be elements of a fuzzy algebra which respect the extended algebra operations under inclusion of fuzzy subsets. The family of fuzzy subalgebras of an algebra is an algebraic closure system in I×. Thus the set of fuzzy subalgebras is a complete lattice. A fuzzy equivalence relation on a set is defined and a partition of such a relation into a class of fuzzy subsets is derived. Using these ideas, fuzzy functions between sets, fuzzy congruence relations, and fuzzy homomorphisms are defined. The kernels of fuzzy homomorphisms are proved to be fuzzy congruence relations, paving the way for the fuzzy isomorphism theorem. Finally, we sketch some ideas on free fuzzy subalgebras and polynomial algebras. In a nutshell, we can say that this thesis treats the central ideas of universal algebras, namely subalgebras, homomorphisms, equivalence and congruence relations, isomorphism theorems and free algebra in the fuzzy set theory setting

Thesis (Ph.D.)--York University, 2008. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 275-278). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR45986

In this thesis two problems concerning linear transformations are discussed. Both problems involve linear transformations which are not in some sense semi-simple; otherwise they are unrelated. In part I we present a proof of the theorem that a linear transformation, of a finite dimensional vector space over a field, which has the property that the irreducible factors of its minimal polynomial are separable is the sum of a semi-simple linear transformation and a nilpotent linear transformation, which commute with the original transformation and are polynomials in the original transformation.. We present an example to show that such a decomposition is not always possible. In parts II and III we obtain some representation theorems for closed algebras of linear transformations on a Banach space which are generated by spectral operators. Since such an algebra is the direct sum of its radical and a space of continuous functions its radical can be investigated more readily than the radical of an arbitrary non-semi- simple commutative Banach algebra. In part II we remark that the reduction theory for rings of operators allows one to reduce the problem of representing a spectral operator, T, on a Hilbert space to the problem of representing a quasi-nilpotent operator. When T is of type m+1 and has a "simple" spectrum it is quite easy to obtain an explicit representation of T. In part III we consider spectral operators on a Banach space. We impose quite stringent conditions, hoping that the theorems obtained for these special cases will serve as a model for more general theorems. The knowledge obtained at least delimits the possibilities. We assume that T is of type m+1 and has a "simple" spectrum. One other condition, which is satisfied if the space, X, on which T acts is separable, is imposed. We are then able to obtain a representation of X as a function space. These function spaces are modelled on the analogy of the Orlicz spaces. We are also able to obtain a representation of the not necessarily semi-simple algebra generated by T and its associated projections as an algebra of functions. / Science, Faculty of / Mathematics, Department of / Graduate

This dissertation deals with the classification of varieties according to their Mal'cev properties. In general the so called Mal'cev-type theorems illustrate an interplay between first order properties of a given class of algebras and the lattice properties of the congruence lattices of algebras of the considered class. CHAPTER 1. A survey of some notational conventions, relevant definitions and auxiliary results is presented. Several examples of less frequently used algebras are given together with the important properties of some of them. The term algebra T(X) and useful results concerning 'term' operations are established. A K-reflection is defined and a connection between a K-reflection of an algebra and whether a class K satisfies an identity of the algebra is established. CHAPTER 2. The Mal'cev-type theorems are presented in complete detail for varieties which are congruence permutable, congruence distributive, arithmetical, congruence modular and congruence regular. Several examples of varieties which exhibit these properties are presented together with the necessary verifications. CHAPTER 3. A general scheme of algorithmic character for some Mal'cev conditions is presented. R. Wille (1970) and A. F. Pixley (1972) provided algorithms for the classification of varieties which exhibit strong Mal'cev properties. This chapter is largely devoted to a modification of the Wille-Pixley schemes. It must be noted that this modification is quite different from all such published schemes. The results are the same as in Wille's scheme but slightly less general than in Pixley's. The text presented here, however is much simpler. As an example, the scheme is used to confirm Mal'cev's original theorem on congruence permutable varieties. Finally, the so-called Chinese var£ety is defined and Mal'cev conditions are established for such a variety of algebras . CHAPTER 4. A comprehensive survey of literature concerning Mal'cev conditions is given in this chapter. / Thesis (M.Sc.)-University of Natal, Durban, 1991.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, Dec. 2004. / Title from title screen. Includes bibliographical references.

Thesis (doctoral)--Universität Bonn, 1986. / Includes bibliographical references (p. 141-147).