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 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. Ch. 1. Definitions and examples -- ch. 2. How to represent a poset -- ch. 3. Poset morphisms -- ch. 4. Construction of new posets from old posets -- ch. 5. Connectedness -- ch. 6. Linear extensions Partially ordered sets (posets) have a long history beginning with the first recognition of ordering in the integers.
