Spelling suggestions: "subject:"iniversal algebra"" "subject:"niversal algebra""
1 |
Contributions to the theory of pre-BCK-algebrasSpinks, Matthew (Matthew James), 1970- January 2002 (has links)
Abstract not available
|
2 |
Results on the Computational Complexity of Linear Idempotent Mal'cev ConditionsHorowitz, Jonah 04 1900 (has links)
<p>In this thesis we examine the computational complexity of determining the satisfaction of various Mal'cev conditions. First we present a novel classification of linear idempotent Mal'cev conditions based on the form of the equations with which they are represented. Using this classification we present a class of conditions which can be detected in polynomial time when examining idempotent algebras. Next we generalize an existing result of Freese and Valeriote by presenting another class of conditions whose satisfaction is exponential time hard to detect in the general case, and en route we prove that it is equally hard to detect local constant terms. The final new contribution is an extension of a recent result of Maróti to a subclass class of weak Mal'cev conditions, proving that their detection is decidable and providing a rough upperbound for the complexity of the provided algorithm for said detection. We close the thesis by reviewing the current state of knowledge with respect to determining satisfaction of linear idempotent Mal'cev conditions.</p> / Doctor of Philosophy (PhD)
|
3 |
Some Mal'cev conditions for varieties of algebras.Moses, Mogambery. January 1991 (has links)
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.
|
4 |
Problém splnitelnosti omezení a univerzální algebra / Constraint Satisfaction Problem and Universal AlgebraKazda, Alexandr January 2013 (has links)
The thesis consists of a collection of my contributions to universal algebra. Motivated by the Constraint Satisfaction Problem (CSP), we study the algebras of polymorphisms of relational structures. We begin by showing by an algebraic argument (and a bit of calculus) that random relational structures' CSP is almost always NP-complete. We then study digraphs with a Maltsev polymorphism, and conclude that such digraphs must also have a majority polymorphism. Next, we show how to use absorption tech- niques to prove that congruence modular reflexive digraphs must have an NU operation. We close our work by giving an algebraic proof of a result (first obtained by graph theorists) that 3-conservative relational structures with only unary and binary relations either define NP-complete CSP, or CSP for them can be solved by the local consistency algorithm. 1
|
5 |
A Finiteness Criterion for Partially Ordered Semigroups and its Applications to Universal AlgebraNelson, Evelyn M. 05 1900 (has links)
<p> A finiteness criterion is given for finitely generated positively ordered semigroups and this is used to show that various semigroups of operators in universal algebra are finite.</p> / Thesis / Master of Science (MSc)
|
6 |
The algebra of entanglement and the geometry of compositionHadzihasanovic, Amar January 2017 (has links)
String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of higher algebraic theories, and as combinatorial descriptions of "directed spaces". Operations of polygraphs modelled on operations of topological spaces are used as the foundation of a compositional universal algebra, where sliding moves arise from tensor products of polygraphs. We reconstruct several higher algebraic theories in this framework. In this regard, the standard formalism of polygraphs has some technical problems. We propose a notion of regular polygraph, barring cell boundaries that are not homeomorphic to a disk of the appropriate dimension. We define a category of non-degenerate shapes, and show how to calculate their tensor products. Then, we introduce a notion of weak unit to recover weakly degenerate boundaries in low dimensions, and prove that the existence of weak units is equivalent to a representability property. We then turn to applications of diagrammatic algebra to quantum theory. We re-evaluate the category of Hilbert spaces from the perspective of categorical universal algebra, which leads to a bicategorical refinement. Then, we focus on the axiomatics of fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits. The ZW calculus has several advantages over ZX calculi, including a computationally meaningful normal form, and a fragment whose diagrams can be read as setups of fermionic oscillators. Moreover, its generators reflect an operational classification of entangled states of 3 qubits. We conclude with generalisations of the ZW calculus to higher-dimensional systems, including the definition of a universal set of generators in each dimension.
|
7 |
Completely regular semiringsSchumann, Rick 16 July 2013 (has links) (PDF)
Vollständig reguläre Halbgruppen weisen eine stark regelmäßige Struktur auf, die verschiedenste Zerlegungsmöglichkeiten gestatten. Ziel dieser Dissertation ist es, diese strukturelle Regelmäßigkeit auf Halbringe zu übertragen und die gewonnenen Algebren zu untersuchen. Mehrere Charakterisierungen werden herausgearbeitet, aufgrund derer es sich herausstellt, dass die Klasse aller vollständig regulären Halbringe eine Varietät bilden, deren Untervarietäten in der Folge untersucht werden. Zentrale Bedeutung haben dabei vollständig einfache Halbringe, deren Analyse einen der Schwerpunkte der Arbeit darstellt. Es zeigt sich, dass diese Bausteine vollständig regulärer Halbringe untereinander eine feste Struktur besitzen, selber aber auch als Zusammensetzung von isomorphen Halbringen aufgefasst werden können. Außerdem werden orthodoxe Halbringe, also Halbringe, deren idempotente Elemente einen Unterhalbring bilden, betrachtet. Zunächst wird dabei wieder auf mehrere Teilklassen eingegangen, bevor abschließend für beliebige vollständig reguläre Halbringe eine Beschreibung der kleinsten Kongruenz angegeben wird, deren Faktorhalbring orthodox ist.
|
8 |
Varieties of residuated latticesGalatos, Nikolaos. January 1900 (has links)
Thesis (Ph. D. in Mathematics)--Vanderbilt University, 2003. / Title from PDF title screen. Includes bibliographical references and index.
|
9 |
Residually small varieties and commutator theory.Swart, Istine Rodseth. January 2000 (has links)
Chapter 0
In this introductory chapter, certain notational and terminological conventions
are established and a summary given of background results that are
needed in subsequent chapters.
Chapter 1
In this chapter, the notion of a "weak conguence formula" [Tay72], [BB75] is
introduced and used to characterize both subdirectly irreducible algebras and
essential extensions. Special attention is paid to the role they play in varieties
with definable principal congruences.
The chapter focuses on residually small varieties; several of its results take
their motivation from the so-called "Quackenbush Problem" and the "RS Conjecture".
One of the main results presented gives nine equivalent characterizations
of a residually small variety; it is largely due to W. Taylor. It is followed
by several illustrative examples of residually small varieties.
The connections between residual smallness and several other (mostly categorical)
properties are also considered, e.g., absolute retracts, injectivity, congruence
extensibility, transferability of injections and the existence of injective
hulls. A result of Taylor that establishes a bound on the size of an injective
hull is included.
Chapter 2
Beginning with a proof of A. Day's Mal'cev-style characterization of congruence
modular varieties [Day69] (incorporating H.-P. Gumm's "Shifting Lemma"),
this chapter is a self-contained development of commutator theory in
such varieties. We adopt the purely algebraic approach of R. Freese and R.
McKenzie [FM87] but show that, in modular varieties, their notion of the commutator
[α,β] of two congruences α and β of an algebra coincides with that
introduced earlier by J. Hagemann and C. Herrmann [HH79] as well as with
the geometric approach proposed by Gumm [Gum80a],[Gum83].
Basic properties of the commutator are established, such as that it behaves
very well with respect to homomorphisms and sufficiently well in products
and subalgebras. Various characterizations of the condition "(x, y) Є [α,β]”
are proved. These results will be applied in the following chapters. We show
how the theory manifests itself in groups (where it gives the familiar group
theoretic commutator), rings, modules and congruence distributive varieties.
Chapter 3
We define Abelian congruences, and Abelian and affine algebras. Abelian
algebras are algebras A in which [A2, A2] = idA (where A2 and idA are the
greatest and least congruences of A). We show that an affine algebra is polynomially
equivalent to a module over a ring (and is Abelian). We give a proof that
an Abelian algebra in a modular variety is affine; this is Herrmann's Funda-
mental Theorem of Abelian Algebras [Her79]. Herrmann and Gumm [Gum78],
[Gum80a] established that any modular variety has a so-called ternary "difference
term" (a key ingredient of the Fundamental Theorem's proof). We derive
some properties of such a term, the most significant being that its existence
characterizes modular varieties.
Chapter 4
An important result in this chapter (which is due to several authors) is the
description of subdirectly irreducible algebras in a congruence modular variety.
In the case of congruence distributive varieties, this theorem specializes to
Jόnsson's Theorem.
We consider some properties of a commutator identity (Cl) which is a necessary
condition for a modular variety to be residually small. In the main
result of the chapter we see that for a finite algebra A in a modular variety,
the variety V(A) is residually small if and only if the subalgebras of A satisfy
(Cl). This theorem of Freese and McKenzie also proves that a finitely generated
congruence modular residually small variety has a finite residual bound,
and it describes such a bound. Thus, within modular varieties, it proves the
RS Conjecture.
Conclusion
The conclusion is a brief survey of further important results about residually
small varieties, and includes mention of the recently disproved (general) RS
Conjecture. / Thesis (M.Sc.)-University of Natal, Durban, 2000.
|
10 |
Conformal densities and deformations of uniform loewner metric spaces /Ruth, Harry Leonard, Jr. January 2008 (has links)
Thesis (Ph.D.)--University of Cincinnati, 2008. / Committee/Advisors: David Herron PhD (Committee Chair), David Minda PhD (Committee Member), Nageswari Shanmugalingam PhD (Committee Member). Includes bibliographical references and abstract.
|
Page generated in 0.1776 seconds