Basic Posets

In this book, the author announces the class of problems called "entropy of knots" and gives the overview of existing topological invariants. He constructs statistical models on braids using the representations of Alexander and Jones invariants and puts forward the question of limit distribution of these invariants for randomly generated braids. The relation of highest powers of corresponding algebraic invariants to the Lyapunov exponents of the products of noncommunicative matrices is shown. Also the problem of conditional joint limit distribution for "brownian bridges" on braids is discussed. Special cases of noncommutative groups PSL(2,R), PSL(2,Z) and braid groups are considered in detail. In the volume, the author also discusses the application of conformal methods for the explicit construction of topological invariants for random walks on multiconnected manifolds. Furthermore the connection of these topological invariants and the monodromy properties of correlation functions of some conformal theories are also discussed Definitions and examples; how to represent a poset; poset morphisms; construction of new posets from old posets; connectedness; linear extensions Preface Chapter 1 Definitions and Examples Chapter 2 How to Represent a Poset 2.1 Hasse Diagram 2.2 Labeling with a Hasse Diagram 2.3 The Adjacency Matrix of a Hasse Diagram 2.4 Interval Order and Semiorder 2.5 Angle Order and Circle Order 2.6 Other Representations 2.7 Order Geometry Chapter 3 Poset Morphisms Chapter 4 Construction of New Posets from Old Posets 4.1 Sum and Ordinal Sum 4.2 Product Order and Lexicographic Order 4.3 Exponential Posets Chapter 5 Connectedness Chapter 6 Linear Extensions 6.1 Linear Extensions 6.2 Representation Polynomials and Linear Extensions 6.3 Dimension 6.4 Applications of Linear Extensions Appendix References Notations and Symbols Index This book introduces the reader to the general theory of partially ordered sets, i.e., posets. The text is presented in a rather informal manner, with interesting examples and computations, which rely on the Hasse diagram to build graphical intuition for the structure of finite posets. The proofs of a small number of theorems is included in the appendix. Important examples especially the Letter N poset, which plays a role akin to that of the Petersen graph in providing a candidate counterexample to many propositions, are used repeatedly throughout the text. This book introduces the reader to the general theory of partially ordered sets, i.e., posets. The text is presented in a rather informal manner, with interesting examples and computations, which rely on the Hassle diagram to build graphical intuition for the structure of finite posits. The proofs of a small number of theorems is included in the appendix. Important examples especially the Letter N poset, which plays a role akin to that of the Petersen graph in providing a candidate counterexample to many propositions, are used repeatedly throughout the text An introduction to the theory of partially-ordered sets, or "posets". The text is presented in rather an informal manner, with examples and computations, which rely on the Hasse diagram to build graphical intuition for the structure of infinite posets. The proofs of a small number of theorems is included in the appendix. Important examples, especially the Letter N poset, which plays a role akin to that of the Petersen graph in providing a candidate counterexample to many propositions, are used repeatedly throughout the text. Partially ordered sets (posets) have a long history beginning with the first recognition of ordering in the integers.