On the Existence of Right Adjoints for Surjective Mappings between Fuzzy Structures

Abstract. We continue our study of the characterization of existence of adjunctions (isotone Galois connections) whose codomain is insufficiently structured. This paper focuses on the fuzzy case in which we have a fuzzy ordering ρA on A and a surjective mapping f : A, ≈A → B, ≈B compatible with respect to the fuzzy equivalences ≈A and ≈B. Specifically, the problem is to find a fuzzy ordering ρB and a compatible mapping g : B, ≈B → A, ≈A such that the pair (f, g) is a fuzzy adjunction

Classification and homological invariants of compact quantum groups of combinatorial type

Compact quantum groups can be found by solving certain combinatorics problems, as first shown by Banica and Speicher. Any system of partitions of finite sets which is closed under reflection and two kinds of concatenation gives rise to a quantum subgroup of the free orthogonal quantum group. Later Freslon, Tarrago and Weber extended this construction to colored partitions. Only recently, Mančinska and Roberson generalized this from finite sets to finite graphs. The present thesis contributes to the classification programs for quantum groups induced by two-colored partitions in Chapter 1 and those induced by uncolored graphs in Chapter 2. While these constructions produce numerous quantum groups, little is known about which of those are actually new and not isomorphic to others. In an effort to elucidate this, Chapter 3 shows that any such quantum group interpolating the unitary group and the free unitary quantum group can be written as a quotient of a wreath graph product of one of the two. Another way of making distinctions between such quantum groups of combinatorial type is to study quantum group invariants, such as cohomology. Chapter 4 computes the first order with trivial coefficients for the discrete duals of all of Tarrago and Weber’s quantum groups. For a handful of those Chapter 5 computes the L²-Betti numbers following Bichon, Kyed and Raum’s method. Chapter 6 proposes a common categorial framework covering all the aforementioned constructions for the first time.Durch das Lösen gewisser Kombinatorikrätsel lassen sich kompakte Quantengruppen finden, wie von Banica und Speicher gezeigt. Jede Sammlung von Partitionen endlicher Mengen, die unter Spiegelung und zwei Arten Konkatenierung abgeschlossen ist, ergibt eine Unterquantengruppe der freien orthogonalen Quantengruppe. Freslon, Tarrago und Weber erweiterten dies auf “gefärbte Partionen”. Erst kürzlich ersetzten Mancinska und Roberson die endlichen Mengen durch endliche Graphen. Die Dissertation trägt zu zwei entsprechenden Klassifikationsvorhaben bei: zweifarbige Partitionen in Kapitel 1, ungefärbte Graphen in Kapitel 2. Zwar ergeben sich viele Quantengruppen. Doch ist nur wenig darüber bekannt, welche davon tatsächlich neu sind. Um dieser Frage nachzugehen, wird in Kapitel 3 bewiesen, dass jede solche Quantengruppe zwischen der unitären Gruppe und der freien unitären Quantengruppe Quotient eines Kranzgraphprodukts einer dieser beiden ist. Eine andere Möglichkeit, solche Quantengruppen kombinatorischen Typs von einander zu unterscheiden bieten Invarianten wie Kohomologie. Von letzterer, mit trivialen Koeffizienten, wird in Kapitel 4 die erste Ordnung berechnet, und zwar für die diskreten Dualen aller von Tarrago und Webers Quantengruppen. Für eine handvoll davon werden in Kapitel 5 noch nach der Methode von Bichon, Kyed und Raum die L2-Betti-Zahlen bestimmt. Kapitel 6 enthält den Vorschlag eines gemeinsamen Rahmens für erstmals alle zuvor genannten Konstruktionen von Quantengruppen.Deutsche Forschungsgemeinschaft, SFB-TRR 195, IRTG scholarshi

Reconstruction of functions from minors

The central notion of this thesis is the minor relation on functions of several arguments. A function f: A^n→B is called a minor of another function g: A^m→B if f can be obtained from g by permutation of arguments, identification of arguments, and introduction of inessential arguments. We first provide some general background and context to this work by presenting a brief survey of basic facts and results concerning different aspects of the minor relation, placing some emphasis on the author’s contributions to the field. The notions of functions of several arguments and minors give immediately rise to the following reconstruction problem: Is a function f: A^n→B uniquely determined, up to permutation of arguments, by its identification minors, i.e., the minors obtained by identifying a pair of arguments? We review known results – both positive and negative – about the reconstructibility of functions from identification minors, and we outline the main ideas of the proofs, which often amount to formulating and solving reconstruction problems for other kinds of mathematical objects. We then turn our attention to functions determined by the order of first occurrence, and we are interested in the reconstructibility of such functions. One of the main results of this thesis states that the class of functions determined by the order of first occurrence is weakly reconstructible. Some reconstructible subclasses are identified; in particular, pseudo-Boolean functions determined by the order of first occurrence are reconstructible. As our main tool, we introduce the notion of minor of permutation. This is a quotient-like construction for permutations that parallels minors of functions and has some similarities to permutation patterns. We develop the theory of minors of permutations, focusing on Galois connections induced by the minor relation and on the interplay between permutation groups and minors of permutations. Our results will then find applications in the analysis of the reconstruction problem of functions determined by the order of first occurrence

An Abstract Algebraic Theory of L-Fuzzy Relations for Relational Databases

Classical relational databases lack proper ways to manage certain real-world situations including imprecise or uncertain data. Fuzzy databases overcome this limitation by allowing each entry in the table to be a fuzzy set where each element of the corresponding domain is assigned a membership degree from the real interval [0…1]. But this fuzzy mechanism becomes inappropriate in modelling scenarios where data might be incomparable. Therefore, we become interested in further generalization of fuzzy database into L-fuzzy database. In such a database, the characteristic function for a fuzzy set maps to an arbitrary complete Brouwerian lattice L. From the query language perspectives, the language of fuzzy database, FSQL extends the regular Structured Query Language (SQL) by adding fuzzy specific constructions. In addition to that, L-fuzzy query language LFSQL introduces appropriate linguistic operations to define and manipulate inexact data in an L-fuzzy database. This research mainly focuses on defining the semantics of LFSQL. However, it requires an abstract algebraic theory which can be used to prove all the properties of, and operations on, L-fuzzy relations. In our study, we show that the theory of arrow categories forms a suitable framework for that. Therefore, we define the semantics of LFSQL in the abstract notion of an arrow category. In addition, we implement the operations of L-fuzzy relations in Haskell and develop a parser that translates algebraic expressions into our implementation