Characterizations of Planar Lattices Christian Zschalig by Left-relations Umschlaggestaltung: Juliane Trinckauf Christian Zschalig Dissertation Characterizations of Planar Lattices by Left-relations Characterizations of Planar Lattices by Left-relations Dissertation zur Erlangung des akademischen Grades Doktor rerum naturalium (Dr. rer. nat.) vorgelegt der Fakult¨at Mathematik und Naturwissenschaften der Technischen Universit¨at Dresden von Dipl.-Math. Christian Zschalig geboren am 07.01.1975 in Riesa Gutachter: Prof. Bernhard Ganter Prof. Ulrike Baumann Prof. Bernd Sohroeder Eingereicht am: 08.10.2008 Disputation am: 05.02.2009 2 Danksagung Ganz herzlich m¨ochte ich mich bei allen bedanken, die mich – in welcher Art auch immer – unterstutzt¨ haben, diese Dissertation schreiben zu k¨onnen. Zuallererst geht mein gr¨oßter Dank an meine Freundin Susi Muller.¨ Vor allem in der letzten, hektischen Phase der Dissertation hat sie mir die Kraft und Ruhe gegeben, die Arbeit vollenden zu k¨onnen. Es ist wunderbar, dass jeder meiner Tage durch einen Sonnenstrahl erhellt wird. Ein herzlicher Dank gebuhrt¨ meinem Betreuer Bernhard Ganter; fur¨ die Frei- heit, die er mir w¨ahrend der Promotionsphase ließ, ein Thema nach meinen Vorstellungen zu bearbeiten und fur¨ die Liebe zur Algebra, die er in mir ge- weckt hat. Immer eine große Hilfe im universit¨aren Alltag war mir Ulrike Baumann. Sie ermutigte mich, mit der Arbeit als Tutor zu beginnen und unterstutzte¨ mich in all den Jahren bei meiner Lehrt¨atigkeit. Ohne sie h¨atte ich mit der Promotion sicher nicht begonnen. Meine ersten wissenschaftlichen Gehversuche unternahm ich meist zusammen mit Heiko Reppe. Vielen Dank ihm fur¨ das viele Korrekturlesen und die ge- meinsamen Konferenzbesuche. Ich hoffe, ich konnte ihn in ¨ahnlicher Weise un- terstutzen¨ wie er mich. Allen noch nicht namentlich genannten Mitgliedern des Instituts fur¨ Algebra sei fur¨ die warmherzige und unverkrampfte Atmosph¨are gedankt, durch die Spaß und Freude an Forschung und Lehre in mir geweckt und erhalten wurden. Vielen Dank an Jojo Klein, Phillip Olk, Kerstin Knott, Sanni Kehr, Ronny Peters, Ina Schaarschmidt und alle anderen von der Bierstubencrew“ fur¨ die ” vielen Kaffees und die entspannenden Gespr¨ache, die ich mit euch genießen konnte. Fur¨ die hervorragende Umschlaggestaltung schulde ich Juliane Trinckauf großen Dank. Zum Schluß, aber keinesfalls am wenigsten m¨ochte ich meinen Eltern Klaus und Maria Zschalig danken, die meine mathematischen Neigungen schon von klein- auf gef¨ordert haben und die mir jederzeit jegliche Unterstutzung¨ zukommen ließen. 4 Contents 1 Introduction 7 2 Motivation 8 2.1 WhyDoWeNeedNiceDiagrams? . 9 2.2 WhatIsaNiceDiagram? . .. .. 11 2.3 WhyPlanarity?........................... 12 3 Preliminaries 14 3.1 BasicDefinitions .......................... 15 3.2 DiagramsofOrderedStructures . 19 3.3 PlanarLattices ........................... 23 3.4 PlaneDiagrams........................... 29 3.5 FormalConceptAnalysis. 32 3.6 Ferrers-relations. 34 4 Left-relations on Lattices 37 4.1 DefinitionandBasicProperties . 38 4.2 Left-relationsandConjugateOrders. 41 4.3 PlanarityConditions . 44 4.4 PlanarContexts........................... 50 5 Left-relations on Diagrams 58 5.1 Definition .............................. 59 5.2 Coherence toLeft-relationsonLattices . 62 5.3 HowtoDrawPlaneDiagrams . 67 5.3.1 AttributeAdditiveDiagrams. 68 5.3.2 LayerDiagrams . .. .. 70 5.3.3 Layered Attribute Additive Diagrams . 72 6 Left-relations on Contexts 73 6.1 DefinitionandFirstProperties. 74 6.2 TheComponentsofFerrers-graphs . 79 6 CONTENTS 6.2.1 ConnectionsinIntervals . 81 6.2.2 Induced Relations on Components of Γ . 85 6.2.3 Edge Sequences in Γ and Connections in V ........ 92 6.2.4 ComponentsofType1andType2 . 98 6.3 Gaining Left-orders out of Ferrers-graphs . 104 6.3.1 TurningComponentsofΓ . 104 6.3.2 The Number of Plane Diagrams of a Lattice . 107 6.4 Determining All Non-similar Plane Diagrams of a Lattice . 113 7 Conclusion and Further Work 116 7.1 Conclusion..............................116 7.2 FurtherWork ............................118 7.2.1 Algebraic Consequences . 118 7.2.2 Minimizing the Crossing Number . 118 7.2.3 DrawingNiceDiagrams . 120 Chapter 1 Introduction In the last thirty years, the development of lattice theory was boosted by the newly emerging Formal Concept Analysis (FCA). Caused by its application- oriented objective, several issues gained a greater importance. In particular, the interest in nicely readable automatically layouted diagrams grew. This work is a contribution to this concern. We particularly investigate planar lattices, namely their characterization and representation by diagrams. Thereby, our considerations are driven by the existence of an additional order sorting the elements of a planar lattice (or poset) from left to right in contrast to the generic one which leads top-down. All structures considered will be finite. We will describe the role of FCA as a model for handling information in Chapter 2. Lattice diagrams can be used for visualizing knowledge. We will shortly explain what a “nice” diagram may constitute and why planarity plays such an important role. Chapter 3 will lay the mathematical foundation of our work. We will recall the basic concepts of order theory, the notion of a diagram and important results about planar lattices. Furthermore we introduce FCA and Ferrers-graphs of contexts (a basic FCA structure determining a lattice). Afterwards, we will introduce left-relations on lattices in Chapter 4. We will analyze its connections with conjugate orders, standard tools for describing planarity. Furthermore, we will give possibilities to handle left-relations more efficiently and to characterize planar contexts. In the following Chapter 5 we investigate how to actually draw a planar lattice without edge crossings. Quite helpful is the fact that left-relations on lattices can be topologically found in the shape of their respective diagrams. In the penultimate Chapter 6 we char- acterize planar lattices by a property of the Ferrers-graph of its context. We will give an algorithm that finds all “topologically different” plane diagrams. Finally, we will summarize the achieved results in Chapter 7. We will give further aims seemingly interesting within the influence of this work and some hints how to approach them. Chapter 2 Motivation A lattice (see Definition 3.7) is planar (see Definition 3.16) if it can be drawn in the plane without edge crossings. In this work we want to investigate this property in detail with the help of so called left-relations. However, in this chapter we want to illustrate why we are concerned with planar lattices at all. Of course, that issue is of interest just from an algebraic point of view; planarity itself and its connection to the dimension (see Definitions 3.5, 3.6) of a lattice are fruitful research areas (see Theorem 3.20). In fact, our work was motivated rather from an application-oriented view. By the use of Formal Concept Analysis (see Section 3.5) we realize that lattice diagrams can impart information. In order to be accepted by a human user, these diagrams must look “nice” and easily readable. Although it is a hard task to mathematize human esthetic sensations, we may assume that in most cases reducing the number of edge crossings improves the diagram’s quality. In particular, a lattice should be laid out without edge crossings at all, if possible. We want to explain and reason these statements more precisely in this chapter. In Section 2.1 we will highlight the importance of diagrams for handling pieces of information. In particular, we will give a flavor of the field of Formal Concept Analysis including its philosophical foundation and possible applications for structuring data. In Section 2.2 we will investigate the problems that occur by automatically drawing diagrams of discrete structures. Thereby, we refer to graphs rather than lattices since the quality of their drawings has been analyzed more specif- ically. We will explain the idea of esthetic criteria as a possibility to mathe- matize human esthetic conceptions and the desire of easy readability. Finally we will illustrate in Section 2.3 why we concentrated on the esthetic criterion of minimizing the number of edge crossings and, in particular, on the recognition and graphical design of planar lattices as those structures that allow diagrams without edge crossings at all. 2.1 Why Do We Need Nice Diagrams? 9 2.1 Why Do We Need Nice Diagrams? The rapid development and increased use of computers and the worldwide spread of the internet facilitates the sharing of enormous amounts of informa- tion. However, data can quickly become too complex to organize (retrieve, store, classify, manipulate, . ) it efficiently. A new research area, namely information science is concerned with that issue. Data is often presented in tables. Unfortunately, this is quite an unintuitive way of making the inherent information accessible to a human user. It seems to be useful to process the raw data into a form that fits us better. An ap- propriate method should apply our understanding of the principles of human thinking. A simple and well-accepted philosophical model is that of thinking in concepts. These are considered to be the smallest units of human thoughts. They are related by judgements and conclusions. The field of formal concept analysis (FCA), introduced by R. Wille and B. Ganter in 1982, is founded on the mathematization of the interrelation between data tables (called formal contexts) and respective