(Formula present) Descent classes Dj = (w G W \ l(ws) < l(w) s G /, for all s G S), Ç S, are also analyzed using generalized quotients. Connections with the weak order are explored and it is shown that W/V is always a complete meet-semilattice and a convex order ideal as a subset of W under weak order. For finite groups W, generalized quotients are the same thing as lower intervals in the weak order. The Möbius function on W/V under Bruhat order takes values in (-1, 0, +!). We show that Bruhat intervals in WIV, for general V Ç W, are lexicographically shellable. Such sets WfV, here called generalized quotients are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when V Ç S (i.e., for minimal coset representatives modulo a parabolic subgroup). The readers are challenged to practice those techniques by solving exercises, a list of which concludes each chapter.įor (W, S) a Coxeter group, we study sets of the form W/V = (wew \ l(wv) = l(w) + l(v) for all v G V), where V ÇW. As the central problem of the book may in fact be solved soon, the book aims to go further, providing the readers with many techniques that can be used to answer more general questions. Throughout the investigation, the readers are introduced to a large number of tools in the theory of Coxeter groups, drawn from dozens of recent articles by prominent researchers in geometric and combinatorial group theory, among other fields. Most of the results addressed here concern conditions which can be seen as varying degrees of uniqueness of representations of Coxeter groups. A number of theorems relating to this problem are stated and proven. The primary purpose of the book is to highlight approximations to the difficult isomorphism problem in Coxeter groups. It is self-contained, and accessible even to advanced undergraduate students of mathematics. The book is the first to give a comprehensive overview of the techniques and tools currently being used in the study of combinatorial problems in Coxeter groups. On the other hand, the main innovative aspect of modal logic SOn is the presence of the sequence (◯1,…,◯n), since ◯i establishes whether an agent is interested in knowing a given fact at time ti. Furthermore, the operator □i of (□1,…,□n) represents the knowledge of an agent at time ti, and it coincides with the necessity modal operator of S5 logic. Modal logic SOn is characterized by the sequences (□1,…,□n) and (◯1,…,◯n) of n modal operators corresponding to a sequence (t1,…,tn) of consecutive times. Finally, we provide the original modal logicSOn with semantics based on sequences of orthopairs, and we employ it to describe the knowledge of an agent that increases over time, as new information is provided. Moreover, as an application, we show that a sequence of orthopairs can be used to represent an examiner’s opinion on a number of candidates applying for a job, and we show that opinions of two or more examiners can be combined using operations between sequences of orthopairs in order to get a final decision on each candidate. Also, we prove several representation theorems representing the class of finite centered Kleene algebras with the interpolation property, and some classes of finite residuated lattices (more precisely, we consider Nelson algebras, Nelson lattices, IUML-algebras and Kleene lattice with implication ) as sequences of orthopairs.
Mainly, our aim is to introduce some operations between sequences of orthopairs, and to discover how to generate them starting from the operations concerning standard rough sets (defined in ). In this thesis, a generalization of the classical Rough set theory is developed considering the so-called sequences of orthopairs that we define in as special sequences of rough sets. We discuss the issue with this previous work. This paper also identifies an error in a long-standing solution to the problem of representing lattices. This structure is much faster for lattices with low-degree elements.
We investigate dual spaces of congruence lattices of algebras in a congruence-distributive variety V\documentclass)$ time, where $d$ is the maximum degree of any element in the transitive reduction graph of the lattice.