This thesis consists of four papers about the intersection between semigroup theory, category theory and representation theory. We say that a representation of a semigroup by a matrix semigroup is effective if it is injective and define the effective dimension of a semigroup S as the minimal n such that S has an effective representation by square matrices of size n. A multisemigroup is a generalization of a semigroup where the multiplication is set-valued, but still associative. A 2-category consists of objects, 1-morphisms and 2-morphisms. A finitary 2-category has finite dimensional vector spaces as objects and linear maps as morphisms. This setting permits the notion of indecomposable 1-morphisms, which turn out to form a multisemigroup. Paper I computes the effective dimension Hecke-Kiselman monoids of type A. Hecke-Kiselman monoids are defined by generators and relations, where the generators are vertices and the relations depend on arrows in a given quiver. Paper II computes the effective dimension of path semigroups and truncated path semigroups. A path semigroup is defined as the set of all paths in a quiver, with concatenation as multiplication. It is said to be truncated if we introduce the relation that all paths of length N are zero. Paper III defines the notion of a multisemigroup with multiplicities and discusses how it better captures the structure of a 2-category, compared to a multisemigroup (without multiplicities). Paper IV gives an example of a family of 2-categories in which the multisemigroup with multiplicities is not a semigroup, but where the multiplicities are either 0 or 1. We describe these multisemigroups combinatorially.

Doctor of Philosophy / Department of Mathematics / Yan S. Soibelman / The motivic Donaldson-Thomas theory of 2-dimensional Calabi-Yau categories can be induced from the theory of 3-dimensional Calabi-Yau categories via dimensional reduction. The cohomological Hall algebra is one approach to the motivic Donaldson-Thomas invariants. Given an arbitrary quiver one can construct a double quiver, which induces the preprojective algebra. This corresponds to a 2-dimensional Calabi-Yau category. One can further construct a triple quiver with potential, which gives rise to a 3-dimensional Calabi-Yau category. The critical cohomological Hall algebra (critical COHA for short) is defined for a quiver with potential. Via the dimensional reduction we obtain the cohomological Hall algebra (COHA for short) of the preprojective algebra. We prove that a subalgebra of this COHA consists of a semicanonical basis, thus is related to the generalized quantum groups. Another approach is motivic Hall algebra, from which an integration map to the quantum torus is constructed. Furthermore, a conjecture concerning some invariants of 2-dimensional Calabi-Yau categories is made. We investigate the correspondence between the A∞-equivalent classes of ind-constructible 2-dimensional Calabi-Yau categories with a collection of generators and a certain type of quivers. This implies that such an ind-constructible category can be canonically reconstructed from its full subcategory consisting of the collection of generators.

This thesis deals with the topic of category management in online retailing. Its goal is to identify the assortment factors that affect a company's financial performance. The result of the thesis work is a model applicable in strategic decision making on assortment. The first part consists of basic concepts of retail, marketing and strategic management. The purpose of this part is to provide the reader with the theoretical background for the following practical part. The research for the practical part is conducted on the basis of interviews with experts in the field and the author's own research in business. Each chapter presents a group of metrics that are based on analysis of business data in online retail market. Metrics describe the specific features of a product category and their impact on a company's revenue or costs. Each metric comes also with an assessment of the given category based on business data. The aim of the thesis is to give a comprehensive overview of assortment decision making. The thesis should also help the reader gain basic knowledge useful for managing an online retail store.

We investigate categorical models of the unit-free multiplicative and multiplicative-additive fragments of linear logic by representing derivations as particular structures known as dinatural transformations. Suitable categories are considered to satisfy a property known as full completeness if all such entities are the interpretation of a correct derivation. It is demonstrated that certain Hyland-Schalk double glueings [HS03] are capable of transforming large numbers of degenerate models into more accurate ones. Compact closed categories with finite biproducts possess enough structure that their morphisms can be described as forms of linear arrays. We introduce the notion of an extended tensor (or ‘extensor’) over arbitrary semirings, and show that they uniquely describe arrows between objects generated freely from the tensor unit in such categories. It is made evident that the concept may be extended yet further to provide meaningful decompositions of more general arrows. We demonstrate how the calculus of extensors makes it possible to examine the combinatorics of certain double glueing constructions. From this we show that the Hyland-Tan version [Tan97], when applied to compact closed categories satisfying a far weaker version of full completeness, produces genuine fully complete models of unit-free multiplicative linear logic. Research towards the development of a full completeness result for the multiplicative-additive fragment is detailed. The proofs work for categories of finite arrays over certain semirings under both the Hyland-Tan and Schalk [Sch04] constructions. We offer a possible route to finishing this proof. An interpretation of these results with respect to linear logic proof theory is provided, and possible further research paths and generalisations are discussed.

The concept of a recursive function has been extensively studied using traditional tools of computability theory. However, with the development of category-theoretic methods it has become possible to study recursion in a more general (abstract) sense. The particular model this thesis is structured around is known as a Turing category. The structure within a Turing category models the notion of partiality as well as recursive computation, and equips us with the tools of category theory to study these concepts. The goal of this work is to build a formal language description of this computation model. Specifically, to use the Coq proof assistant to formulate informal definitions, propositions and proofs pertaining to Turing categories in the underlying formal language of Coq, the Calculus of Co-inductive Constructions (CIC). Furthermore, we have instantiated the more general Turing category formalism with a CIC description of the category which models the language of partial recursive functions exactly.

Cette thèse examine comment les catégories de marché -ensembles qui partagent des similarités cognitives et culturelles- impactent la performance et l'évaluation des entreprises.Le consensus répandu dans la littérature indique que les organisations qui évoluent dans plus d'une catégorie sont sanctionnées tant au plan économique que social.Remettant en cause ce consensus actuel sur "l'impératif catégorique", cette thèse avance l'idée que les acteurs d'un marché ont un rôle plus complexe que simplement réprimer toute violation des catégories établies. Aussi dans ce contexte, être engagée dans plusieurs catégories de marché pour une organisation à la fois améliore son évaluation sociale mais réduit sa performance en cas de perceptions divergentes de ses affiliations catégorielles. Les données empiriques de cette thèse portent sur les cabinets d'avocats d'affaires dans trois grandes métropoles (New-York, Paris et Londres) au cours d'une décennie (2000-2010). Les cabinets d'avocats multi-services - ceux qui exercent dans plusieurs domaines du droit- obtiennent une meilleure évaluation de la part des clients tant au niveau global du cabinet que pour chacune de leurs spécialités juridiques. Toutefois, les désaccords entre clients en terme d'évaluation portant sur chacune des spécialités juridiques offertes détériorent la performance financière des cabinets. Cette thèse approfondit notre compréhension du rôle que jouent les catégories sur les marchés et les stratégies d'extensions catégorielles que les entreprises mettent en oeuvre. Ce travail contribue également aux études sur le champ juridique et a des implications pour la conduite stratégique des cabinets d'avocats d'affaires / This dissertation explores how market categories - clusters that share cognitive and cultural similarities - impact firms' performance and evaluation. Pervasive consensus in literature indicates organizations that do not fall into a single category suffer economic and social disadvantages.Unsettling this current consensus about the categorical imperative, this dissertation advocates that external audiences have a more complex role than simply patrolling the boundaries and sanctioning any infringement of established categories. They scrutinize categories in various ways depending on their needs. They infer some characteristics of firms from one category membership to another. They diverge about the category memberships and evaluation of firms. This dissertation provides evidence that in such cases, spanning categories both leads to positive social evaluations for organizations, but decreases performance in case of inconsistency across categories. Empirically I study the corporate legal services market in three major financial locations (New-York City, Paris and London) over a decade (2000-2010). My findings are twofold:(i) multi-category law firms- those that are engaged in several practice areas of law - receive better social evaluation from clients both at the firm level and at the practice area level; (ii) disagreement among clients' evaluation about law firms' practice areas undermines their financial performance. This dissertation deepens our understanding of the role that categorical structures play in markets and the category-stretching strategies firms implement to better navigate the "category map". This work contributes also to research in legal studies and has implications for law firms' business development

In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function 𝑓 which is the uniform limit of a sequence of sawtooth functions 𝑓ₙ : [0, 1] → [0, 1] with increasingly sharp teeth. Let 𝑋 = [0, 1] x [0, 1] and 𝐹(𝑋) be the Hausdorff metric space determined by 𝑋. We define contraction maps 𝑤₁ , 𝑤₂ , 𝑤₃ on 𝑋. These maps define a contraction map 𝑤 on 𝐹(𝑋) via 𝑤(𝐴) = 𝑤₁(𝐴) ⋃ 𝑤₂(𝐴) ⋃ 𝑤₃(𝐴). The iteration under 𝑤 of the diagonal in 𝑋 defines a sequence of graphs of continuous functions 𝑓ₙ. Since 𝑤 is a contraction map in the compact metric space 𝐹(𝑋), 𝑤 has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function 𝑓. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in 𝐶[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set of proper local extrema is residual in 𝐶[0,1]. In the fourth and last chapter we actually construct our function and prove it is continuous, nowhere-differentiable and has a dense set of proper local extrema. Lastly we iterate the set {(0,0), (1,1)} under 𝑤 and plot its points. Any terms not defined in Chapters 2 through 4 may be found in [2,4]. The same applies to the basic properties of metric spaces which have not been explicitly stated. Throughout, we will let 𝒩 and 𝕽 denote the natural numbers and the real numbers, respectively.

The Master thesis discusses the origin, development and termination of Alfred Radok's foundation. Furthermore the thesis deals with Alfred Radok's award endowment fund, which was focusing on the presentation of Alfred Radok's award. The aim of this thesis is to assess the activities of the foundation and of the endowment fund with a special focus on issues Alfred Radok's awards ceremony during its 23-year's operation. A great benefit of the thesis is information which is not going to be accessible anymore due to the end of Alfred Radok's awards. Alfred Radok's awards were composed of two parts. First part was an anonymous dramatic competition which awards the best script of an original Czech theatre play. And the second part was awarding of a production categories. Results of the production categories were based on a theatre critic's survey. The aim was to find out how are the critics and publicists satisfied with this survey. The method of the research was qualitative investigation based on semi-structured interviews. In conclusion, the data was evaluated and interpreted.

We consider right modules over a ring, as models of a first order theory. We explorethe definable sets and the definable bijections between them. We employ the notionsof Euler characteristic and Grothendieck ring for a first order structure, introduced byJ. Krajicek and T. Scanlon in [24]. The Grothendieck ring is an algebraic structurethat captures certain properties of a model and its category of definable sets.If M is a module over a product of rings A and B, then M has a decomposition into a direct sum of an A-module and a B-module. Theorem 3.5.1 states that then the Grothendieck ring of M is the tensor product of the Grothendieck rings of the summands.Theorem 4.3.1 states that the Grothendieck ring of every infinite module over afield or skew field is isomorphic to Z[X].Proposition 5.2.4 states that for an elementary extension of models of anytheory, the elementary embedding induces an embedding of the corresponding Grothendieck rings. Theorem 5.3.1 is that for an elementary embedding of modules, we have the stronger result that the embedding induces an isomorphism of Grothendieck rings.We define a model-theoretic Grothendieck ring of the category Mod-R and explorethe relationship between this ring and the Grothendieck rings of general right R-modules. The category of pp-imaginaries, shown by K. Burke in [7] to be equivalentto the subcategory of finitely presented functors in (mod-R; Ab), provides a functorial approach to studying the generators of theGrothendieck rings of R-modules. It is shown in Theorem 6.3.5 that whenever R andS are Morita equivalent rings, the rings Grothendieck rings of the module categories Mod-R and Mod-S are isomorphic.Combining results from previous chapters, we derive Theorem 7.2.1 saying that theGrothendieck ring of any module over a semisimple ring is isomorphic to a polynomialring Z[X1,...,Xn] for some n.

This diploma thesis deals with the analysis of the actual situation of cereal drinks category, focusing on the Caro brand, and uses marketing research oriented to public and their perception to cereal drinks category and Caro brand. Based on this, it formulates results and recommendations that can be useful for potential help in the area of the brand marketing activities. The diploma thesis is divided into two parts. First part focuses on theoretical and methodic fundamentals, such as marketing and its specific characteristics related to the grocery, marketing environment, marketing mix and the concept of marketing research. Second part deals with the Caro brand marketing, cereal drinks category, analysis of the actual offer situation and marketing research, which is focused on the public perception of cereal drinks category and brand awareness. Based on the ascertained results of analysis and marketing research, recommendations are formulated with respect to brand marketing activities.