Chapter 5 Partial Orders, Lattices, Well Founded Orderings
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On the Lattice Structure of Quantum Logic
BULL. AUSTRAL. MATH. SOC. MOS 8106, *8IOI, 0242 VOL. I (1969), 333-340 On the lattice structure of quantum logic P. D. Finch A weak logical structure is defined as a set of boolean propositional logics in which one can define common operations of negation and implication. The set union of the boolean components of a weak logical structure is a logic of propositions which is an orthocomplemented poset, where orthocomplementation is interpreted as negation and the partial order as implication. It is shown that if one can define on this logic an operation of logical conjunction which has certain plausible properties, then the logic has the structure of an orthomodular lattice. Conversely, if the logic is an orthomodular lattice then the conjunction operation may be defined on it. 1. Introduction The axiomatic development of non-relativistic quantum mechanics leads to a quantum logic which has the structure of an orthomodular poset. Such a structure can be derived from physical considerations in a number of ways, for example, as in Gunson [7], Mackey [77], Piron [72], Varadarajan [73] and Zierler [74]. Mackey [77] has given heuristic arguments indicating that this quantum logic is, in fact, not just a poset but a lattice and that, in particular, it is isomorphic to the lattice of closed subspaces of a separable infinite dimensional Hilbert space. If one assumes that the quantum logic does have the structure of a lattice, and not just that of a poset, it is not difficult to ascertain what sort of further assumptions lead to a "coordinatisation" of the logic as the lattice of closed subspaces of Hilbert space, details will be found in Jauch [8], Piron [72], Varadarajan [73] and Zierler [74], Received 13 May 1969. -
Relations on Semigroups
International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Relations on Semigroups 1D.D.Padma Priya, 2G.Shobhalatha, 3U.Nagireddy, 4R.Bhuvana Vijaya 1 Sr.Assistant Professor, Department of Mathematics, New Horizon College Of Engineering, Bangalore, India, Research scholar, Department of Mathematics, JNTUA- Anantapuram [email protected] 2Professor, Department of Mathematics, SKU-Anantapuram, India, [email protected] 3Assistant Professor, Rayalaseema University, Kurnool, India, [email protected] 4Associate Professor, Department of Mathematics, JNTUA- Anantapuram, India, [email protected] Abstract: Equivalence relations play a vital role in the study of quotient structures of different algebraic structures. Semigroups being one of the algebraic structures are sets with associative binary operation defined on them. Semigroup theory is one of such subject to determine and analyze equivalence relations in the sense that it could be easily understood. This paper contains the quotient structures of semigroups by extending equivalence relations as congruences. We define different types of relations on the semigroups and prove they are equivalence, partial order, congruence or weakly separative congruence relations. Keywords: Semigroup, binary relation, Equivalence and congruence relations. I. INTRODUCTION [1,2,3 and 4] Algebraic structures play a prominent role in mathematics with wide range of applications in science and engineering. A semigroup -
Scott Spaces and the Dcpo Category
SCOTT SPACES AND THE DCPO CATEGORY JORDAN BROWN Abstract. Directed-complete partial orders (dcpo’s) arise often in the study of λ-calculus. Here we investigate certain properties of dcpo’s and the Scott spaces they induce. We introduce a new construction which allows for the canonical extension of a partial order to a dcpo and give a proof that the dcpo introduced by Zhao, Xi, and Chen is well-filtered. Contents 1. Introduction 1 2. General Definitions and the Finite Case 2 3. Connectedness of Scott Spaces 5 4. The Categorical Structure of DCPO 6 5. Suprema and the Waybelow Relation 7 6. Hofmann-Mislove Theorem 9 7. Ordinal-Based DCPOs 11 8. Acknowledgments 13 References 13 1. Introduction Directed-complete partially ordered sets (dcpo’s) often arise in the study of λ-calculus. Namely, they are often used to construct models for λ theories. There are several versions of the λ-calculus, all of which attempt to describe the ‘computable’ functions. The first robust descriptions of λ-calculus appeared around the same time as the definition of Turing machines, and Turing’s paper introducing computing machines includes a proof that his computable functions are precisely the λ-definable ones [5] [8]. Though we do not address the λ-calculus directly here, an exposition of certain λ theories and the construction of Scott space models for them can be found in [1]. In these models, computable functions correspond to continuous functions with respect to the Scott topology. It is thus with an eye to the application of topological tools in the study of computability that we investigate the Scott topology. -
Topological Duality and Lattice Expansions Part I: a Topological Construction of Canonical Extensions
TOPOLOGICAL DUALITY AND LATTICE EXPANSIONS PART I: A TOPOLOGICAL CONSTRUCTION OF CANONICAL EXTENSIONS M. ANDREW MOSHIER AND PETER JIPSEN 1. INTRODUCTION The two main objectives of this paper are (a) to prove topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. The paper is first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper. Regarding objective (a), consider the following simple question: Is there a subcategory of Top that is dually equivalent to Lat? Here, Top is the category of topological spaces and continuous maps and Lat is the category of bounded lattices and lattice homomorphisms. To date, the question has been answered positively either by specializing Lat or by generalizing Top. The earliest examples are of the former sort. Tarski [Tar29] (treated in English, e.g., in [BD74]) showed that every complete atomic Boolean lattice is represented by a powerset. Taking some historical license, we can say this result shows that the category of complete atomic Boolean lattices with complete lat- tice homomorphisms is dually equivalent to the category of discrete topological spaces. Birkhoff [Bir37] showed that every finite distributive lattice is represented by the lower sets of a finite partial order. Again, we can now say that this shows that the category of finite distributive lattices is dually equivalent to the category of finite T0 spaces and con- tinuous maps. -
Large but Finite
Department of Mathematics MathClub@WMU Michigan Epsilon Chapter of Pi Mu Epsilon Large but Finite Drake Olejniczak Department of Mathematics, WMU What is the biggest number you can imagine? Did you think of a million? a billion? a septillion? Perhaps you remembered back to the time you heard the term googol or its show-off of an older brother googolplex. Or maybe you thought of infinity. No. Surely, `infinity' is cheating. Large, finite numbers, truly monstrous numbers, arise in several areas of mathematics as well as in astronomy and computer science. For instance, Euclid defined perfect numbers as positive integers that are equal to the sum of their proper divisors. He is also credited with the discovery of the first four perfect numbers: 6, 28, 496, 8128. It may be surprising to hear that the ninth perfect number already has 37 digits, or that the thirteenth perfect number vastly exceeds the number of particles in the observable universe. However, the sequence of perfect numbers pales in comparison to some other sequences in terms of growth rate. In this talk, we will explore examples of large numbers such as Graham's number, Rayo's number, and the limits of the universe. As well, we will encounter some fast-growing sequences and functions such as the TREE sequence, the busy beaver function, and the Ackermann function. Through this, we will uncover a structure on which to compare these concepts and, hopefully, gain a sense of what it means for a number to be truly large. 4 p.m. Friday October 12 6625 Everett Tower, WMU Main Campus All are welcome! http://www.wmich.edu/mathclub/ Questions? Contact Patrick Bennett ([email protected]) . -
Nieuw Archief Voor Wiskunde
Nieuw Archief voor Wiskunde Boekbespreking Kevin Broughan Equivalents of the Riemann Hypothesis Volume 1: Arithmetic Equivalents Cambridge University Press, 2017 xx + 325 p., prijs £ 99.99 ISBN 9781107197046 Kevin Broughan Equivalents of the Riemann Hypothesis Volume 2: Analytic Equivalents Cambridge University Press, 2017 xix + 491 p., prijs £ 120.00 ISBN 9781107197121 Reviewed by Pieter Moree These two volumes give a survey of conjectures equivalent to the ber theorem says that r()x asymptotically behaves as xx/log . That Riemann Hypothesis (RH). The first volume deals largely with state- is a much weaker statement and is equivalent with there being no ments of an arithmetic nature, while the second part considers zeta zeros on the line v = 1. That there are no zeros with v > 1 is more analytic equivalents. a consequence of the prime product identity for g()s . The Riemann zeta function, is defined by It would go too far here to discuss all chapters and I will limit 3 myself to some chapters that are either close to my mathematical 1 (1) g()s = / s , expertise or those discussing some of the most famous RH equiv- n = 1 n alences. Most of the criteria have their own chapter devoted to with si=+v t a complex number having real part v > 1 . It is easily them, Chapter 10 has various criteria that are discussed more brief- seen to converge for such s. By analytic continuation the Riemann ly. A nice example is Redheffer’s criterion. It states that RH holds zeta function can be uniquely defined for all s ! 1. -
Lattice-Ordered Loops and Quasigroupsl
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOIJRNALOFALGEBRA 16, 218-226(1970) Lattice-Ordered Loops and Quasigroupsl TREVOREVANS Matkentatics Department, Emory University, Atlanta, Georgia 30322 Communicated by R. H, Brwk Received April 16, 1969 In studying the effect of an order on non-associativesystems such as loops or quasigroups, a natural question to ask is whether some order condition which implies commutativity in the group case implies associativity in the corresponding loop case. For example, a well-known theorem (Birkhoff, [1]) concerning lattice ordered groups statesthat if the descendingchain condition holds for the positive elements,then the 1.0. group is actually a direct product of infinite cyclic groups with its partial order induced in the usual way by the linear order in the factors. It is easy to show (Zelinski, [6]) that a fully- ordered loop satisfying the descendingchain condition on positive elements is actually an infinite cyclic group. In this paper we generalize this result and Birkhoff’s result, by showing that any lattice-ordered loop with descending chain condition on its positive elements is associative. Hence, any 1.0. loop with d.c.c. on its positive elements is a free abelian group. More generally, any lattice-ordered quasigroup in which bounded chains are finite, is isotopic to a free abelian group. These results solve a problem in Birkhoff’s Lattice Theory, 3rd ed. The proof uses only elementary properties of loops and lattices. 1. LATTICE ORDERED LOOPS We will write loops additively with neutral element0. -
Binary Relations and Functions
Harvard University, Math 101, Spring 2015 Binary relations and Functions 1 Binary Relations Intuitively, a binary relation is a rule to pair elements of a sets A to element of a set B. When two elements a 2 A is in a relation to an element b 2 B we write a R b . Since the order is relevant, we can completely characterize a relation R by the set of ordered pairs (a; b) such that a R b. This motivates the following formal definition: Definition A binary relation between two sets A and B is a subset of the Cartesian product A×B. In other words, a binary relation is an element of P(A × B). A binary relation on A is a subset of P(A2). It is useful to introduce the notions of domain and range of a binary relation R from a set A to a set B: • Dom(R) = fx 2 A : 9 y 2 B xRyg Ran(R) = fy 2 B : 9 x 2 A : xRyg. 2 Properties of a relation on a set Given a binary relation R on a set A, we have the following definitions: • R is transitive iff 8x; y; z 2 A :(xRy and yRz) =) xRz: • R is reflexive iff 8x 2 A : x Rx • R is irreflexive iff 8x 2 A : :(xRx) • R is symmetric iff 8x; y 2 A : xRy =) yRx • R is asymmetric iff 8x; y 2 A : xRy =):(yRx): 1 • R is antisymmetric iff 8x; y 2 A :(xRy and yRx) =) x = y: In a given set A, we can always define one special relation called the identity relation. -
The Modal Logic of Potential Infinity, with an Application to Free Choice
The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity. -
Some Integral Inequalities for Operator Monotonic Functions on Hilbert Spaces Received May 6, 2020; Accepted June 24, 2020
Spec. Matrices 2020; 8:172–180 Research Article Open Access Silvestru Sever Dragomir* Some integral inequalities for operator monotonic functions on Hilbert spaces https://doi.org/10.1515/spma-2020-0108 Received May 6, 2020; accepted June 24, 2020 Abstract: Let f be an operator monotonic function on I and A, B 2 SAI (H) , the class of all selfadjoint op- erators with spectra in I. Assume that p : [0, 1] ! R is non-decreasing on [0, 1]. In this paper we obtained, among others, that for A ≤ B and f an operator monotonic function on I, Z1 Z1 Z1 0 ≤ p (t) f ((1 − t) A + tB) dt − p (t) dt f ((1 − t) A + tB) dt 0 0 0 1 ≤ [p (1) − p (0)] [f (B) − f (A)] 4 in the operator order. Several other similar inequalities for either p or f is dierentiable, are also provided. Applications for power function and logarithm are given as well. Keywords: Operator monotonic functions, Integral inequalities, Čebyšev inequality, Grüss inequality, Os- trowski inequality MSC: 47A63, 26D15, 26D10. 1 Introduction Consider a complex Hilbert space (H, h·, ·i). An operator T is said to be positive (denoted by T ≥ 0) if hTx, xi ≥ 0 for all x 2 H and also an operator T is said to be strictly positive (denoted by T > 0) if T is positive and invertible. A real valued continuous function f (t) on (0, ∞) is said to be operator monotone if f (A) ≥ f (B) holds for any A ≥ B > 0. In 1934, K. Löwner [7] had given a denitive characterization of operator monotone functions as follows: Theorem 1. -
Advanced Discrete Mathematics Mm-504 &
1 ADVANCED DISCRETE MATHEMATICS M.A./M.Sc. Mathematics (Final) MM-504 & 505 (Option-P3) Directorate of Distance Education Maharshi Dayanand University ROHTAK – 124 001 2 Copyright © 2004, Maharshi Dayanand University, ROHTAK All Rights Reserved. No part of this publication may be reproduced or stored in a retrieval system or transmitted in any form or by any means; electronic, mechanical, photocopying, recording or otherwise, without the written permission of the copyright holder. Maharshi Dayanand University ROHTAK – 124 001 Developed & Produced by EXCEL BOOKS PVT. LTD., A-45 Naraina, Phase 1, New Delhi-110 028 3 Contents UNIT 1: Logic, Semigroups & Monoids and Lattices 5 Part A: Logic Part B: Semigroups & Monoids Part C: Lattices UNIT 2: Boolean Algebra 84 UNIT 3: Graph Theory 119 UNIT 4: Computability Theory 202 UNIT 5: Languages and Grammars 231 4 M.A./M.Sc. Mathematics (Final) ADVANCED DISCRETE MATHEMATICS MM- 504 & 505 (P3) Max. Marks : 100 Time : 3 Hours Note: Question paper will consist of three sections. Section I consisting of one question with ten parts covering whole of the syllabus of 2 marks each shall be compulsory. From Section II, 10 questions to be set selecting two questions from each unit. The candidate will be required to attempt any seven questions each of five marks. Section III, five questions to be set, one from each unit. The candidate will be required to attempt any three questions each of fifteen marks. Unit I Formal Logic: Statement, Symbolic representation, totologies, quantifiers, pradicates and validity, propositional logic. Semigroups and Monoids: Definitions and examples of semigroups and monoids (including those pertaining to concentration operations). -
Distributive Envelopes and Topological Duality for Lattices Via Canonical Extensions
DISTRIBUTIVE ENVELOPES AND TOPOLOGICAL DUALITY FOR LATTICES VIA CANONICAL EXTENSIONS MAI GEHRKE AND SAM VAN GOOL Abstract. We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the stan- dard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. In the course of the paper, we obtain the following results: (1) canoni- cal extensions of bounded lattices are the algebraic versions of the existing dualities for bounded lattices by Urquhart and Hartung, (2) there is a univer- sal construction which associates to an arbitrary lattice two distributive lattice envelopes with an adjoint pair between them, and (3) we give a topological du- ality for bounded lattices which generalizes Priestley duality and which shows precisely which maps between bounded lattices admit functional duals. For the result in (1), we rely on previous work of Gehrke, J´onssonand Harding. For the universal construction in (2), we modify a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of `admis- sibility' to the finitary case. For (3), we use Priestley duality for distributive lattices and our universal characterization of the distributive envelopes. 1. Introduction Topological duality for Boolean algebras [22] and distributive lattices [23] is a useful tool for studying relational semantics for propositional logics. Canonical extensions [16, 17, 11, 10] provide a way of looking at these semantics algebraically. In the absence of a satisfactory topological duality, canonical extensions have been used [3] to treat relational semantics for substructural logics.