Wong, Tak Pui. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 60-62). / Abstracts in English and Chinese. / Table of Contents --- p.ii / Abstract --- p.iii / Acknowledgements --- p.iv / Chapter 0 --- Introduction --- p.1 / Chapter 1 --- Definitions of Plane Tropical Curves --- p.4 / Chapter 1.1 --- Motivation --- p.4 / Chapter 1.2 --- As Varieties over the Field of Puiseux Series --- p.7 / Chapter 1.3 --- As Varieties over the Tropical Semiring --- p.10 / Chapter 1.4 --- A Combinatorial Description of Plane Tropical Curves --- p.16 / Chapter 2 --- Properties of Plane Tropical Curves --- p.21 / Chapter 2.1 --- The Degree-genus Formula --- p.21 / Chapter 2.2 --- Bezout's Theorem --- p.24 / Chapter 3 --- Non-Archimedean Amoebas and Patchworking Method --- p.28 / Chapter 3.1 --- Computing Amoebas --- p.28 / Chapter 3.2 --- Patchworking Method --- p.37 / Chapter 3.2.1 --- Maslov's Dequantization --- p.37 / Chapter 3.2.2 --- Patchworking Method --- p.39 / Chapter 4 --- Mikhalkin's Correspondence Theorem --- p.41 / Chapter 4.1 --- Parameterized Plane Tropical Curves --- p.41 / Chapter 4.2 --- Statement of the Main Theorem --- p.43 / Chapter 4.3 --- Lattice Paths --- p.46 / Chapter 4.4 --- Complex Tropical Curves --- p.51 / Chapter 4.5 --- Sketch Proof of the Main Theorem --- p.54 / Chapter 4.5.1 --- Proof of Lemma 1 --- p.55 / Chapter 4.5.2 --- Proof of Lemma 2 --- p.56 / Bibliography --- p.60

Sign pattern theory examines what can be said about a matrix if one knows the signs of all or some of its entries but not the exact values. Since all we know is the sign of each entry, we can write these sign patterns as matrices whose entries come from the set {+1, -1, 0, #}, where # is used for an unknown sign. Semirings satisfy all properties of rings with unity except the existence of additive inverses. The set {+1, -1, 0, #} can be viewed as a commutative semiring in natural way. In the thesis, we give a semiring version of the Cayley-Dickson construction which allows one to construct the sign pattern semiring from the Boolean semiring. We use tools from Boolean matrices to study sign nonsingular (SNS) matrices. We also investigate different notions of rank of matrices over semirings. For these rank functions we simplify proofs of classical inequalities for the sum and the product of matrices using the semiring versions of the Cauchy-Binet and Laplace theorems. For matrices over the sign pattern semiring, the minimum rank of the sign pattern is compared with the other versions of the rank. We also characterize irreducible powerful sign pattern matrices and investigate the period and base of an SNS matrix.

No description available.

A well known fact of classical analysis is that the Stone- Cech compacti cation of a Tychono space is achieved as the structure space of the commutative unitary ring of real valued continuous functions de ned on it. An extension of this result is obtained [1] using semiring of non-negative real valued continuous functions on the Tychono space. The present work makes a survey of this paper and attempts to extend this to the point free domain, in which it is shown that the collection of frame homomorphisms from the frame of non-negative reals to a frame is a semiring again. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

We investigate commutative semirings, which are formed by a ground set equipped with two binary associative and commutative operations such that one distributes over the other. We narrow down our interest to ideal-simple semirings, that is, semirings without proper ideals. We present the classification of ideal-simple semirings and deal with some classes of ideal-simple semirings, namely semifields and parasemifields. The main result of this thesis is giving tight bounds on the minimal number of generators needed to generate a parasemifield as a semiring. We also study how the semifields that are finitely generated as a semiring look like. Last, but not least, we show that every finitely generated ideal-simple semiring is finitely-generated as a multiplicative semigroup.

Vectors and matrices over the Boolean (0,1) semiring have been studied extensively along with their applications to graph theory. The Boolean (0,1) semiring has been generalized to a class of semirings called chain semirings. This class includes the fuzzy interval. Vectors and matrices over chain semirings are examined. Rank-1 sets of vectors are defined and characterized. These rank-1 sets of vectors are then used to construct spaces of matrices (rank-1 spaces) with the property that all nonzero matrices in the space have semiring rank equal to 1. Finally, three classes of maximal (relative to containment) rank-1 spaces are identified.

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.

We characterized the group of linear operators that strongly preserve r-potent matrices over the binary Boolean semiring, nonbinary Boolean semirings, and zero-divisor free antinegative semirings. We extended these results to show that linear operators that strongly preserve r-potent matrices are equivalent to those linear operators that strongly preserve the matrix polynomial equation p(X) = X. where p(X) = Xr1 + Xr2 + ... + Xrt and r1>r2>...>rt≥2. In addition, we characterized the group of linear operators that strongly preserve r-cyclic matrices over the same semirings. We also extended these results to linear operators that strongly preserve the matrix polynomial equation p(X) = I where p(X) is as above. Chapters I and II of this thesis contain background material and summaries of the work done by other researchers on the linear preserver problem. Characterizations of linear operators in chapters III, IV, V, and VI of this thesis are new.

Reed-Solomon codes are a well known family of error-correcting codes with many good properties. However, they require a finite field to operate, limiting the alphabet size to a prime power. In this work, we build a weaker algebraic structure which supports alphabet of any integer size and requires only standard addition, multiplication and division to implement. Then we study a family of error-correcting codes based on matrix multiplication over this structure. We also adapt the Reed-Solomon code principle on this code family and study its properties. We prove and verify experimentally that while a random code of this family has high distance, the Reed-Solomon adaptation fails to perform well. 1

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.