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.
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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. In fact, one of the results in that paper could be interpreted as a definability theorem for infinitary logic, using topological rather than Boolean valued models.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. The construction of this space is closely related to the one in. The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. In particular, we have added a first section in which we review the basic definitions concerning With the most elementary properties of model theory. Our presentation is entirely selfcontained, and only requires familiarity More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T-provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula of L.
Our theorem states that, if one allows Boolean valued models, the converse holds. The same is of course true for subsets of M" defined To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S M (i.e., a subset S = defined by some formula phi) is invariant under allĪutomorphisms of. This theorem is very easy to state, and its proof only uses elementary tools. In this paper, we will present a definability theorem for first order logic.
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