Defending the Junk Argument *

Total Page:16

File Type:pdf, Size:1020Kb

Defending the Junk Argument * Defending the Junk Argument* . Introduction Unrestricted composition is one of the most controversial doctrines of classical mereology. An objection has been put forward to the view that mereological composition is necessarily unre- stricted by appealing to the possibility of so-called ‘junky’ worlds. A world is junky if everything in that world is a proper part of something (i.e. 8x9y(x < y) is true).¹ e junk argument, due to Bohn [, , ], proceeds as follows: (i) If a world is junky, then the unrestricted fusion axiom for mereology is false at that world. (ii) Junky worlds are metaphysically possible. (iii) erefore, the unrestricted fusion axiom for mereology is either metaphysically contingent or necessarily false. Much attention has been devoted to the defence of premise (ii). I do not wish to challenge the case for (ii).² However, more recently premise (i) has come under fire. For example, Contessa [] and Spencer [] have each provided independent reasons that ‘universalists’ should dispatch the argument by rejecting premise (i). In this paper, I undertake a defence of premise (i) against a variety of objections. In §, I put forward a new objection to premise (i): Bohn’s defence of it presupposes far too much. In particular, I show that Bohn’s suppositions entail mereological extensionalism; that is, these assumptions rule out the view that objects with the same proper parts may be distinct. Two of the most prominent non-extensional mereologies are introduced: the unsupplemented view, and the mutual parts view. A non-extensionalist, then, might have strong reasons for accepting universalism but rejecting the argument. *is paper is a . Please do not cite without permission. Comments are welcome. ¹is is the converse of the notion of a gunky world in which everything in that world has something as a proper part. ²But see Watson [] for discussion. Bohn’s main defence involves three criteria for metaphysical possibility: con- ceivability, advocacy, and consistency. But if we can conceive of junky worlds, and several prominent philosophers have taken the idea seriously, and there are no logical contradictions lurking, then we are hard pressed to deny the mere possibility of the world being junky. ([], ) For the record, I do not think these three criteria are sufficient for metaphysical possibility, but the matter is too complicated for discussion here. | . In §§–, I show that one can defend premise (i) from a much weaker set of assumptions. Hence variants of the junk argument can be shown to apply to non-extensional mereological systems as well. I address unsupplemented mereologies in § which explicitly reject one of Bohn’s assumptions. In order to avoid begging the question against such views, I show that a variant of the junk argument survives even without appeal to any kind of supplementation. § argues that the mutual parts theorist who accepts unrestricted composition can avoid Bohn’s argument. However, the kind of junk compatible the the mutual parts view is only counterfeit — it satisfies the definition in letter, but not in spirit. A revision in the definition of junk yields the desired result. § takes on Contessa’s objection to premise (i). Contessa contends that those who accept unrestricted composition should only accept the existence of binary sums rather than infinitary fusions. I argue this conception of unrestricted composition is problematic: it is in conflict with the existence of gunk satisfying an intuitive remainder principle. Ithenconsiderapossibleresponse based on the mereology of Bostock [], and suggest that it’s not really a version of unrestricted fusion and so changes the terms of the debate. It may well be the most plausible version of a junky mereology, but it hardly constitutes a counterinstance to premise (i). In§, Iconsiderarecentresponsetopremise(i)bySpencer[]. Spencer’s view is that there is no absolutely unrestricted universal quantifier. So, any statement of the unrestricted fusion axiom will simply not rule out the existence of junky worlds. I argue that in order for this response to provide a counterexample to premise (i), we need some further argument. First, since junky worlds are defined using the universal quantifier, what Spencer’s argument allows appears to be worlds that are inexpressibly junky. But the metaphysical possibility of inexpressibly junky worlds needs an independent defence, and is not given (even implicitly) by the acceptance of premise (ii). Second, we may be simply unable to quantify over all the parts of something, or unable to quantify over all the simples. But neither of these options will yield junky worlds. ird, even if we have a principled reason to think that our quantifier is indefinitely extensible upwards along parthood chains, it’s not clear that the resulting expression of the unrestricted fusion axiom is genuinely unrestricted in the required sense. ese defences of premise (i) show that it is particularly resilient and can be based on extremely minimal assumptions. e upshot, then, is that anyone who wants to reject the junk argument must do so by rejecting premise (ii) instead. | . Analyticity & Extensionalism In his argument for premise (i), Bohn relies on the following assumptions, where < is proper parthood, ≤ improper parthood, and ◦ mereological overlap: Transitivity (x < y ^ y < z) ! x < z Weak Supplementation x < y ! 9z(z ≤ y ^ :z ◦ x)) Weak Supplementation says that if something has a proper part, it must have another part disjoint from the first. In combination with transitivity, this entails the asymmetry and irreflexivity of <. Irreflexivity x 6< x Asymmetry x < y ! y 6< x Bohn’s argument for premise (i) is as follows: [A]ssume universalism is true in all possible worlds and that some possible world w is junky. en universalism is true in w. Consider the plurality of everything there is in w, call it aa. By universal instantiation, 9y(aa compose y); by existential instantiation, aa compose U. Bydefinition[ofjunk],itistruein w that 8x9y(x < y). By universal instantiation, 9y(U < y). But U is composed of everything in w, and hence, by definition, everything in w is a part of it, including itself. But then nothing can have U as a proper part because if so, by [weak] supplementation, there would be something that shares no part with U,andhenceisnotapartof U, which contradicts that everything is a part of U. So, if some possible world is junky, then universalism is not necessarily true. Q.E.D. ([, p. , fn ]) Now, the principles use in the argument are admittedly fairly plausible; however, they are far from uncontroversial.³ In fact, as we will see below, there is a growing number of mereologists who accept unrestricted composition but reject at least one. If that weren’t reason enough to generalise the junk argument, it turns out that these ‘minimal’ principles are enough, when combined with unrestricted composition, to yield full classical extensional mereology. is includes perhaps the ³Indeed, Bohn regards these principles as analytic of parthood. I assume at least some mereological principles without existential import are analytically true, and in virtue of that are necessarily true too. […] Minimal Extensional Mereology is a minimum of mereological necessary truths. is system includes (among other things) the asymmetry and transitivity of proper parthood, as well as a weak supplementation principle. ([], ) I do not think weak supplementation is analytic, but I appreciate more needs to be said. See [XXXXX]. | . most controversial principle of all — extensionality which says that composite objects with the same proper parts are identical.⁴ e argument for this claim is due to Varzi []. Informally, imagine you had a counterexam- ple to extensionality; a statue s and its clay c, suppose, are distinct objects with the same proper parts. By asymmetry, s 6< c, and c 6< s.⁵ By unrestricted fusion, there must be a sum s + c (which is distinct from either s or c, since neither is part of the other). Now, s < s + c; however, there is no part of s + c that is disjoint from s,asanypartof c is also part of s by supposition. is violates weak supplementation. Hence, there can be no such counterexample to extensionality. ere are two ways to avoid Varzi’s argument; as a result, there are two main approaches to mereology without extensionality: the unsupplemented view, and the mutual parts view. Both of these approaches are compatible with unrestricted composition; indeed, many proponents of these views themselves accept it. Since each explicitly rejects one of Bohn’s assumptions, the junk argument begs the question against them. But it does so needlessly, or so I shall argue. Weak Supplementation & Junk e first main non-extensionalist option is the unsupplemented approach which drops the weak supplementation axiom above. e unsupplemented approach has access to a well-developed mereology that includes unrestricted fusion. Here is a candidate list of axioms, where ≤, ◦, and F are defined as above. Asymmetry x < y ! y 6< x Transitivity (x < y ^ y < z) ! x < z Unrestricted Composition 8xx9y F(y, xx). Fusions appearing in the unrestricted composition axiom are defined as follows.⁶ Fusion F(t, xx) iff xx ≤ t ^ 8y(xx ≤ y ! t ≤ y) Here, xx ≤ t means that each x that is among xx is part of t. On this view, proper parthood is still a strict partial order. e major change here is that weak supplementation is explicitly rejected. Some unsupplemented mereologists simply drop the ⁴See Varzi [] to get a flavour of the debate. ⁵For, suppose c < s. Asymmetry implies s 6< c. Hence, s and c are not a counterexample to extensionality. Mutatis mutandis for the supposition that s < c. ⁶is definition of fusion is weaker than the one used in Varzi’s argument. For current purposes, it seems prudent to use the weakest plausible form of the fusion axiom. | . axiom and leave it at that.⁷ Others (primarily Gilmore []) have suggested a replacement axiom to capture the intuition behind weak supplementation without risk of extensionality.⁸ But Bohn’s defence of premise (i) explicitly appeals to weak supplementation.
Recommended publications
  • 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.
    [Show full text]
  • Leibniz and the Necessity of the Best Possible World
    This is a preprint of an article whose final and definitive form will be published in the Australasian Journal of Philosophy; the Australasian Journal of Philosophy is available online at: http://www.tandf.co.uk/journals/. LEIBNIZ AND THE NECESSITY OF THE BEST POSSIBLE WORLD Martin Pickup Leibniz has long faced a challenge about the coherence of the distinction between necessary and contingent truths in his philosophy. In this paper, I propose and examine a new way to save genuine contingency within a Leibnizian framework. I conclude that it succeeds in formally solving the problem, but at unbearable cost. I present Leibniz’s challenge by considering God’s choice of the best possible world (Sect. 2). God necessarily exists and necessarily chooses to actualise the best possible world. The actual world therefore could not be different, for if it were different it would be a distinct and inferior world and hence would not be created. In Sect. 3 I defend Leibniz from this challenge. I argue that while it is necessary for God to choose to create the best possible world, it is not necessary for any world to be the best possible. This is because the criterion for judging perfection can itself be contingent. Different criteria will judge different worlds as the best. Thus it is necessary for God to create the best, but not necessary which is the best. Distinguishing between possible worlds in Leibniz’s sense and in the modern sense allows a fuller exposition of this position. There are worries that can arise with the claim that the criterion of perfection is contingent.
    [Show full text]
  • On Considering a Possible World As Actual
    ON CONSIDERING A POSSIBLE WORLD AS ACTUAL Robert Stalnaker “Hell is paved with primary intensions” English proverb1 1. Introduction In Naming and Necessity2, Saul Kripke presented some striking examples that convinced many philosophers that there are truths that are both necessary and a posteriori, and also truths that are both contingent and a priori. The classic examples of the former are identity statements containing proper names (Hesperus = Phosphorus) and statements about the nature of natural kinds (Gold has atomic number 79). Realistic examples of the second kind of statement are harder to come by - perhaps there are none - but once one sees the idea it is easy to construct artificial examples. The most famous example is based on Gareth Evans’s descriptive name “Julius.”3 “Julius” is, by stipulation, a proper name of the person (whoever he might be) who invented the zip. So the statement “Julius invented the zip” can be known a priori to be true, but since the description is used to fix the reference rather than to give the meaning , of the name, the fact that Julius invented the zip is a contingent fact. Someone other than Julius (the person we in fact named) might have invented the zip, and if some else had invented it instead, it would not have been true that Julius invented the zip. The divergence of the two distinctions was surprising, but the examples are simple and relatively transparent. Kripke’s exposition pointed the way to an abstract description of the phenomena that made it clear, on one level at least, what is going on.
    [Show full text]
  • The Parallel Architecture and Its Place in Cognitive Science
    978–0–19–954400–4 Heine-main-drv Heine-Narrog (Typeset by Spi, Chennai) 645 of 778 April 8, 2009 21:55 chapter 23 .............................................................................................................. THE PARALLEL ARCHITECTURE AND ITS PLACE IN COGNITIVE SCIENCE .............................................................................................................. ray jackendoff It has become fashionable recently to speak of linguistic inquiry as biolinguistics, an attempt to frame questions of linguistic theory in terms of the place of lan- guage in a biological context. The Minimalist Program (Chomsky 1995; 2001)is of course the most prominent stream of research in this paradigm. However, an alternative stream within the paradigm, the Parallel Architecture, has been devel- oping in my own work over the past 30 years; it includes two important sub- components, Conceptual Structure and Simpler Syntax (Jackendoff2002; 2007b; Culicover and Jackendoff 2005). This chapter will show how the Parallel Architec- ture is in many ways a more promising realization of biolinguistic goals than the Minimalist Program and that, more than the Minimalist Program, it is conducive to integration with both the rest of linguistic theory and the rest of cognitive science. 978–0–19–954400–4 Heine-main-drv Heine-Narrog (Typeset by Spi, Chennai) 646 of 778 April 8, 2009 21:55 646 ray jackendoff 23.1 Parallel architectures, broadly conceived .........................................................................................................................................
    [Show full text]
  • Frege and the Logic of Sense and Reference
    FREGE AND THE LOGIC OF SENSE AND REFERENCE Kevin C. Klement Routledge New York & London Published in 2002 by Routledge 29 West 35th Street New York, NY 10001 Published in Great Britain by Routledge 11 New Fetter Lane London EC4P 4EE Routledge is an imprint of the Taylor & Francis Group Printed in the United States of America on acid-free paper. Copyright © 2002 by Kevin C. Klement All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any infomration storage or retrieval system, without permission in writing from the publisher. 10 9 8 7 6 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Klement, Kevin C., 1974– Frege and the logic of sense and reference / by Kevin Klement. p. cm — (Studies in philosophy) Includes bibliographical references and index ISBN 0-415-93790-6 1. Frege, Gottlob, 1848–1925. 2. Sense (Philosophy) 3. Reference (Philosophy) I. Title II. Studies in philosophy (New York, N. Y.) B3245.F24 K54 2001 12'.68'092—dc21 2001048169 Contents Page Preface ix Abbreviations xiii 1. The Need for a Logical Calculus for the Theory of Sinn and Bedeutung 3 Introduction 3 Frege’s Project: Logicism and the Notion of Begriffsschrift 4 The Theory of Sinn and Bedeutung 8 The Limitations of the Begriffsschrift 14 Filling the Gap 21 2. The Logic of the Grundgesetze 25 Logical Language and the Content of Logic 25 Functionality and Predication 28 Quantifiers and Gothic Letters 32 Roman Letters: An Alternative Notation for Generality 38 Value-Ranges and Extensions of Concepts 42 The Syntactic Rules of the Begriffsschrift 44 The Axiomatization of Frege’s System 49 Responses to the Paradox 56 v vi Contents 3.
    [Show full text]
  • Proquest Dissertations
    INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy subm itted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Leaming 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 UMI AN ACCOUNT OF THE JUSTIHCATION OF TESTIMONIAL BELIEFS: A RELIABILIST APPROACH DISSERTATION Presented in Partial Fulfillment of the Requirement for the Degree Doctor of Philosophy in the Graduate School of the The Ohio State University By David Ena. M.A. The Ohio State University 2000 Dissertation Committee Approved by Professor Marshall Swain.
    [Show full text]
  • COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES Patrick F. Smith 1. Introduction in This Paper We Continue the Discussion In
    International Electronic Journal of Algebra Volume 16 (2014) 16-31 COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES Patrick F. Smith Received: 18 October 2013; Revised: 25 May 2014 Communicated by Christian Lomp For my good friend John Clark on his 70th birthday Abstract. We examine the properties of certain mappings between the lattice L(R) of ideals of a commutative ring R and the lattice L(RM) of submodules of an R-module M, in particular considering when these mappings are com- plete homomorphisms of the lattices. We prove that the mapping λ from L(R) to L(RM) defined by λ(B) = BM for every ideal B of R is a complete ho- momorphism if M is a faithful multiplication module. A ring R is semiperfect (respectively, a finite direct sum of chain rings) if and only if this mapping λ : L(R) !L(RM) is a complete homomorphism for every simple (respec- tively, cyclic) R-module M. A Noetherian ring R is an Artinian principal ideal ring if and only if, for every R-module M, the mapping λ : L(R) !L(RM) is a complete homomorphism. Mathematics Subject Classification 2010: 06B23, 06B10, 16D10, 16D80 Keywords: Lattice of ideals, lattice of submodules, multiplication modules, complete lattice, complete homomorphism 1. Introduction In this paper we continue the discussion in [7] concerning mappings, in particular homomorphisms, between the lattice of ideals of a commutative ring and the lattice of submodules of a module over that ring. A lattice L is called complete provided every non-empty subset S has a least upper bound _S and a greatest lower bound ^S.
    [Show full text]
  • Wittgenstein's Conception of the Autonomy of Language and Its Implications for Natural Kinds
    Wittgenstein’s Conception of the Autonomy of Language and its Implications for Natural Kinds George Wrisley [email protected] NOTICE TO BORROWERS In presenting this thesis as partial fulfillment of the requirements for an advanced degree from Georgia State University, I agree that the library of the university will make it available for inspection and circulation in accordance with its regulations governing materials of this type. I agree that permission to quote from, to copy from, or to publish from this thesis may be granted by the author, by the professor under whose direction it was written, or by the Dean of the College of Arts & Sciences. Such quoting, copying or publishing must be solely for scholarly purposes and must not involve potential financial gain. It is understood that any copying from or publication of this thesis that involves potential financial gain will not be allowed without written permission of the author. ________________________________________ All dissertations and theses deposited in the Georgia State University Library may be used only in accordance with the stipulations prescribed by the author in the preceding statement. The author of this dissertation is The director of this dissertation is George Wrisley Dr. Grant Luckhardt 8081 Henderson Court Department of Philosophy Alpharetta, GA 30004 College of Arts & Sciences Wittgenstein’s Conception of the Autonomy of Language And its Implications for Natural Kinds A Thesis Presented in Partial Fulfillment of Requirements for the Degree of Master of Arts in the College of Arts and Sciences Georgia State University 2002 by George Wrisley Committee: ______________________________ Dr. Grant Luckhardt, Chair ______________________________ Dr.
    [Show full text]
  • On Removing Ambiguity in Text Understanding
    Language. Information and Computation (PACLIC12). 18-20 Feb. 1998, 271-282 On Removing Ambiguity in Text Understanding Simin Li, Yukihiro Itoh Dept. of Computer Science, Shizuoka Univ. This paper discusses how to remove a kind of ambiguity in a text understanding system based on simulation. The system simulates some events mentioned in text on a world model and observes the behavior of the model during the simulation. Through these processes, it can recognize the other events mentioned implicitly. However, in case the system infers plural number of possible world, it can't decide which one is consistent with the context. We deal with such ambiguities. The ambiguities, in some cases, can be removed by considering contextual information. To remove the ambiguities in the simulation, we define three heuristics based on the characters of the explanatory descriptions and propose an algorithm of "looking ahead". We implement ,ani experimental system. Accqrding to the ,algorithm, the system finds out the supplementary descriptions in the following sentences and removes the ambiguities by using the contents of the supplementary descriptions. Keyword: Text Understanding, Simulation, Ambiguity, Heuristics. 1. Introduction In this paper, we discuss a method of removing ambiguities which appear in the process of text understanding based on simulation. We have been studying a text understanding system paying attention to the importance of an imagerial world model. In [Itoh,92, Itoh,95], we showed that the ability to simulate the matters described by sentences on the imagerial world model is one of the basic abilities to understand texts. We also proposed a method of implementing an imagerial world model, a method of simulating on the model and a method of observing the model to extract propositional expressions.
    [Show full text]
  • Pseudocomplemented Semilattices, Boolean Algebras, and Compatible Products1
    Journal of Algebra 242, 60᎐91Ž. 2001 doi:10.1006rjabr.2001.8807, available online at http:rrwww.idealibrary.com on Pseudocomplemented Semilattices, Boolean Algebras, and Compatible Products1 Antonio Fernandez´´ Lopez and Marıa ´ Isabel Tocon ´ Barroso View metadata, citationDepartamento and similar papers de Algebra, at core.ac.uk Geometrıa´´ y Topologıa, Facultad de Ciencias, brought to you by CORE Uni¨ersidad de Malaga,´´ 29071 Malaga, Spain provided by Elsevier - Publisher Connector E-mail: [email protected], [email protected] Communicated by Georgia Benkart Received November 19, 1999 DEDICATED TO PROFESSOR J. MARSHALL OSBORN Pseudocomplemented semilattices are studied here from an algebraic point of view, stressing the pivotal role played by the pseudocomplements and the relation- ship between pseudocomplemented semilattices and Boolean algebras. Following the pattern of semiprime ring theory, a notion of Goldie dimension is introduced for complete pseudocomplemented lattices and calculated in terms of maximal uniform elements if they exist in abundance. Products in lattices with 0-element are studied and questions about the existence and uniqueness of compatible products in pseudocomplemented lattices, as well as about the abundance of prime elements in lattices with a compatible product, are discussed. Finally, a Yood decomposition theorem for topological rings is extended to complete pseudocom- plemented lattices. ᮊ 2001 Academic Press Key Words: pseudocomplemented semilattice; Boolean algebra; compatible product. INTRODUCTION A pseudocomplemented semilattice is aŽ. meet semilattice S having a least element 0 and is such that, for each x in S, there exists a largest element x H such that x n x Hs 0. In spite of what the name could suggest, a complemented lattice need not be pseudocomplemented, as is easily seen by considering the lattice of all subspaces of a vector space of dimension greater than one.
    [Show full text]
  • Lattices and the Geometry of Numbers
    Lattices and the Geometry of Numbers Lattices and the Geometry of Numbers Sourangshu Ghosha, aUndergraduate Student ,Department of Civil Engineering , Indian Institute of Technology Kharagpur, West Bengal, India 1. ABSTRACT In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has application on various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis etc. This paper will review all the major developments that have occurred in the field of geometry of numbers. In this paper we shall first give a broad overview of the concept of lattice and then discuss about the geometrical properties it has and its applications. 2. LATTICE Before introducing Minkowski’s theorem we shall first discuss what is a lattice. Definition 1: A lattice 흉 is a subgroup of 푹풏 such that it can be represented as 휏 = 푎1풁 + 푎2풁 + . + 푎푚풁 풏 Here {푎푖} are linearly independent vectors of the space 푹 and 푚 ≤ 푛. Here 풁 is the set of whole numbers. We call these vectors {푎푖} the basis of the lattice. By the definition we can see that a lattice is a subgroup and a free abelian group of rank m, of the vector space 푹풏. The rank and dimension of the lattice is 푚 and 푛 respectively, the lattice will be complete if 푚 = 푛. This definition is not only limited to the vector space 푹풏.
    [Show full text]
  • Lattice Theory
    Chapter 1 Lattice theory 1.1 Partial orders 1.1.1 Binary Relations A binary relation R on a set X is a set of pairs of elements of X. That is, R ⊆ X2. We write xRy as a synonym for (x, y) ∈ R and say that R holds at (x, y). We may also view R as a square matrix of 0’s and 1’s, with rows and columns each indexed by elements of X. Then Rxy = 1 just when xRy. The following attributes of a binary relation R in the left column satisfy the corresponding conditions on the right for all x, y, and z. We abbreviate “xRy and yRz” to “xRyRz”. empty ¬(xRy) reflexive xRx irreflexive ¬(xRx) identity xRy ↔ x = y transitive xRyRz → xRz symmetric xRy → yRx antisymmetric xRyRx → x = y clique xRy For any given X, three of these attributes are each satisfied by exactly one binary relation on X, namely empty, identity, and clique, written respectively ∅, 1X , and KX . As sets of pairs these are respectively the empty set, the set of all pairs (x, x), and the set of all pairs (x, y), for x, y ∈ X. As square X × X matrices these are respectively the matrix of all 0’s, the matrix with 1’s on the leading diagonal and 0’s off the diagonal, and the matrix of all 1’s. Each of these attributes holds of R if and only if it holds of its converse R˘, defined by xRy ↔ yRx˘ . This extends to Boolean combinations of these attributes, those formed using “and,” “or,” and “not,” such as “reflexive and either not transitive or antisymmetric”.
    [Show full text]