Motivated by results of linear algebra over fields, rings and tropical semirings, we present a systematic way to understand the behaviour of matrices with entries in an arbitrary semiring. We focus on three closely related problems concerning the row and column spaces of matrices. This allows us to isolate and extract common properties that hold for different reasons over different semirings, yet also lets us identify which features of linear algebra are specific to particular types of semiring. For instance, the row and column spaces of a matrix over a field are isomorphic to each others' duals, as well as to each other, but over a tropical semiring only the first of these properties holds in general (this in itself is a surprising fact). Instead of being isomorphic, the row space and column space of a tropical matrix are anti-isomorphic in a certain order-theoretic and algebraic sense. The first problem is to describe the kernels of the row and column spaces of a given matrix. These equivalence relations generalise the orthogonal complement of a set of vectors, and the nature of their equivalence classes is entirely dependent upon the kind of semiring in question. The second, Hahn-Banach type, problem is to decide which linear functionals on row and column spaces of matrices have a linear extension. If they all do, the underlying semiring is called exact, and in this case the row and column spaces of any matrix are isomorphic to each others' duals. The final problem is to explain the connection between the row space and column space of each matrix. Our notion of a conjugation on a semiring accounts for the different possibilities in a unified manner, as it guarantees the existence of bijections between row and column spaces and lets us focus on the peculiarities of those bijections. Our main original contribution is the systematic approach described above, but along the way we establish several new results about exactness of semirings. We give sufficient conditions for a subsemiring of an exact semiring to inherit exactness, and we apply these conditions to show that exactness transfers to finite group semirings. We also show that every Boolean ring is exact. This result is interesting because it allows us to construct a ring which is exact (also known as FP-injective) but not self-injective. Finally, we consider exactness for residuated lattices, showing that every involutive residuated lattice is exact. We end by showing that the residuated lattice of subsets of a finite monoid is exact if and only if the monoid is a group.

<p>Poluprsten je algebarska struktura (5, + , &bull;) sa dve binarne operacije u kojoj su&nbsp; (S,+ ) i (5, &bull;) polugrupe i druga je distributivna prema prvoj sa obe strane. U radu su uvedeni pojmovi p-polugrupe kao i p-poluprstena. Kažemo daje polugrupa ( S, + ) p-polugrupa ako (Vz G&nbsp; S)(3yG&nbsp; S)(x+py+x =&nbsp; y,py + x+py = z ). Poluprsten ( S, +.&bull;)zovemo p-poluprsten ako (Vz G&nbsp; S)(3yG&nbsp; S)(x + py + x = y,py + x + py = z,4p z2 = 4pz). Dokazano je da je svaka p-polugrupa pokrivena grupama koje su u potpunosti opisane. Takođe je pokazano da su p-poluprsteni pokriveni pretprsteni-ma. Za p = 4A; + 3&nbsp; (kG&nbsp; N0)ili p paran broj p-polugrupe, odnosno p-poluprsteni su varijeteti.</p> / <p>A semiring (5 ,+ ,-) is an algebric structure with two binary operations in which ( S, + ) and&nbsp; (S,&bull;) are semigroups, and the second operation is two-side dis&shy; tributive with respect to the first one. In the present paper notions of p-semigroup and p-semiring are introduced. We say that a semigroup (S&#39;, + ) is a p-semigroup if (Vx &pound; S)(3y &pound;&nbsp; S)(x + py + x = y,py + x + py = x).A semiring (S&#39;, + , &bull;) is called a p-semiring if (Vx &pound;&nbsp; S)(3y&pound;&nbsp; S)(x +py + x = y,py + x + py = x,4px2 = 4px). It is proved that each p-semigroup is covered by groups which are completely described. It is also proved that p-semirings are covered by prering. For&nbsp; p = 4k + 3 (k &pound; No) or for even p, the class of p-semigroups, respectively of p-semirings are varieties.</p>

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.

We show that the termination of Mohri's algorithm is decidable for polynomially ambiguous weighted finite automata over the tropical semiring which gives a partial answer to a question by Mohri [29]. The proof relies on an improvement of the notion of the twins property and a Burnside type characterization for the finiteness of the set of states produced by Mohri's algorithm.

<p>Integralne nejednakosti Jensena i Čebiševa<br />uopštene su za integrale bazirane na neaditivnim<br />merama. Prvo uopštenje dokazano je za<br />pseudo-integral skupovno-vrednosne&nbsp; funkcije, a<br />drugo za pseudo-integral realno-vrednosne funkcije<br />u odnosu na intervalno-vrednosnu -meru.<br />Dokazana je i uopštena nejednakost Čebiševa<br />za pseudo-integral realno-vrednosne&nbsp; funkcije i<br />njena dva intervalno-vrednosna oblika. Nejednakost<br />Jensena je primenjena u principu premije, a<br />nejednakost Čebiševa na procenu verovatnoće.</p> / <p>Integral inequalities of Jensen and Chebyshev type are<br />generalized for integrals based on nonadditive measures.<br />The first generalization is proven for the pseudointegral<br />of a set valued function and the second one<br />for the pseudo-integral of a real-valued function with<br />respect to the interval-valued -measure. Generalized<br />Chebyshev inequality for the pseudo-integral of a realvalued<br />function and its two interval-valued forms are<br />proven. Jensen inequality is applied in the premium<br />principle and Chebyshev inequality is applied to the<br />probability estimation.</p>

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18041号 / 理博第3919号 / 新制||理||1566(附属図書館) / 30899 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 西村 進, 教授 加藤 毅, 教授 長谷川 真人 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Title: Algebraic Substructures in ℂ Author: Vítězslav Kala Department: Department of Algebra Supervisor: Prof. RNDr. Tomáš Kepka, DrSc., Department of Algebra Abstract: We study the structure of finitely generated semirings, parasemifields and other algebraic structures, developing and applying tools based on the geom- etry of algebraic substructures of the Euclidean space ℂ . To a parasemifield which is finitely generated as a semiring we attach a certain subsemigroup of the semigroup ℕ0 (defined using elements such that + = for some ∈ and ∈ ℕ). Algebraic and geometric properties of carry important structural information about ; we use them to show that if a parasemifield is 2-generated as a semiring, then it is additively idempotent. We also provide a ring-theoretic reformulation of this conjecture in the case of -generated semirings. We also classify all additively idempotent parasemifields which are finitely gen- erated as semirings by using the fact that they correspond to certain finitely generated unital lattice ordered groups. Busaniche, Cabrer, and Mundici [4] re- cently classified these using the combinatorial and geometric notion of a stellar sequence which is a sequences of certain simplicial complexes in [0, 1] . We use their results to prove that each such parasemifield is a finite product of...

Cryptography based on semirings can be one of the possible approaches for the post-quantum cryptography in the public-key schemes. In our work, we are interested in only one concrete semiring - tropical algebra. We are examining one concrete scheme for the key-agreement protocol - tropical Stickel's protocol. Although there was introduced an attack on it, we have implemented this attack and more importantly, stated its complexity. Further, we propose other variants of Stickel's protocol and we are investigating their potential for practical usage. During the process, we came across the theory of tropical matrix powers, thus we want to make an overview of it due to the use in cryptography based on matrices over the tropical algebra semiring. 1

Cette thèse se propose d’étudier des problèmes de classification des groupes quantiques via des invariants issus de leur théorie de représentation. Plus précisément, nous classifions les algèbres de Hopf possédant un semi-anneau de fusion isomorphe à un groupe algébrique réductif donné G. De tels groupes quantiques sont alors appelés G-déformations. Dans cette thèse, nous étudions les cas GL(2) et SO(3). Nous donnons une classification complète des GL(2)-déformations en construisant une famille d’algèbres de Hopf indexées par des matrices inversibles. Nous décrivons leurs catégories de comodules et donnons certains résultats de classification quant à leurs objets de Hopf-Galois. Ensuite, nous donnons une classification des SO(3)-déformations compactes tout en étudiant le cas non-compact. Finalement, la dernière partie de la thèse est une étude de l’algèbre sous-jacente à une certaine famille d’algèbres de Hopf, dont nous exhibons une base. Cette base nous permet de calculer le centre des ces algèbres ainsi que quelques groupes de (co)homologie. / The purpose of this dissertation is to classify quantum groups according to invariants coming from their representation theory. More precisely, we classify Hopf algebras having a fusion semiring isomorphic to that of a given reductive algebraic group G. Such a quantum group is called a G-deformation. We study the case of GL(2) and SO(3). We give a complete classification of GL(2)-deformations by building a family of Hopf algebras parametrized by invertible matrices. We describe their comodule category and we give some classification results about the Hopf-Galois objects. We also classify compact SO(3)-deformations and we study the noncompact case. Finally, the last part of this dissertation is a study of the underlying algebra of some Hopf algebras, for which we exhibit a linear basis. This basis allows us to compute the centre and some (co)homology groups of those algebras.

Die Arbeit beschäftigt sich mit fast-idempotenten Halbringen, die eine Verallgemeinerung der idempotenten Halbringe darstellen. Es werden - ausgehend von Halbringen mit zwei Elementen - bis auf isomorphe Bilder sämtliche fast-idempotente Halbringe mit drei Elementen generiert, diejenigen Halbringe, die schon in durch zweielementige Halbringe erzeugten Varietäten liegen, aussortiert und die in den verbleibenden elf Halbringen gültigen Gleichungen charakterisiert. Der Verband L(IA3) der Varietäten generiert durch fast-idempotente Halbringe mit maximal drei Elementen wird mit Hilfe eines Kontexts mit 21 Halbringen als Attribute und 28 trennenden Gleichungen als Objekte vollständig bestimmt und besteht aus 19.901 Varietäten. Im Anschluss richtet sich der Fokus der Arbeit auf den Verband L(IA) der fast-idempotenten Halbringe. In diesem werden insbesondere die Varietät V = [xy = yx, xy = xy+x] und deren Untervarietäten V_k = [x^k = x^(k+1)], k >= 2; untersucht. Für all diese Varietäten wird jeweils eine Konstruktionsmethode für eine abzählbare Kette an Untervarietäten der gegebenen Varietät eingeführt und somit schließlich gezeigt, dass der Verband L(IA) aus mindestens abzählbar unendlich vielen Varietäten besteht.