DOCSLIB.ORG

The Congruences of a Finite Lattice a Proof-By-Picture Approach

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Congruences of a Finite Lattice a Proof-By-Picture Approach The Congruences of a Finite Lattice A Proof-by-Picture Approach Second Edition George Gr¨atzer arXiv:2104.06539v1 [math.RA] 13 Apr 2021 Birkh¨auserVerlag, New York @ 2016 Birkh¨auserVerlag To L´aszl´oFuchs, my thesis advisor, my teacher, who taught me to set the bar high; and to my coauthors, who helped me raise the bar. Short Contents Table of Notation xiii Picture Gallery xvii Preface to the Second Edition xix Introduction xxi I A Brief Introduction to Lattices 1 1 Basic Concepts 3 2 Special Concepts 19 3 Congruences 35 4 Planar Semimodular Lattices 47 Bibliography 325 ix Contents Table of Notation xiii Picture Gallery xvii Preface to the Second Edition xix Introduction xxi I A Brief Introduction to Lattices 1 1 Basic Concepts 3 1.1 Ordering3 1.1.1 Ordered sets3 1.1.2 Diagrams5 1.1.3 Constructions of ordered sets5 1.1.4 Partitions7 1.2 Lattices and semilattices9 1.2.1 Lattices9 1.2.2 Semilattices and closure systems 10 1.3 Some algebraic concepts 12 1.3.1 Homomorphisms 12 1.3.2 Sublattices 13 1.3.3 Congruences 15 2 Special Concepts 19 2.1 Elements and lattices 19 2.2 Direct and subdirect products 20 2.3 Terms and identities 23 2.4 Gluing 28 xi xii Contents 2.5 Modular and distributive lattices 30 2.5.1 The characterization theorems 30 2.5.2 Finite distributive lattices 31 2.5.3 Finite modular lattices 32 3 Congruences 35 3.1 Congruence spreading 35 3.2 Finite lattices and prime intervals 38 3.3 Congruence-preserving extensions and variants 41 4 Planar Semimodular Lattices 47 4.1 Basic concepts 47 4.2 SPS lattices 48 4.3 Forks 49 4.4 Rectangular lattices 51 Bibliography 325 Glossary of Notation Symbol Explanation Page 0I and 1I the zero and unit of the interval I 14 Atom(U) set of atoms of the ideal U Aut L automorphism group of L 12 Bn boolean lattice with n atoms 4 Cl(L); Cll(L); Cul(L) left boundary chains of a planar lattice 47 Cn n-element chain 4 con(a; b) smallest congruence under which a b 16 con(c) principal congruence for a color c ≡ 41 con(H) smallest congruence collapsing H 16 con(p) principal congruence generated by a prime interval p 39 Con L congruence lattice of L 15 ConJ L ordered set of join-irreducible congruences of L 39 ConM L ordered set of meet-irreducible congruences of L Cr(L); Clr(L); Cur(L) right boundary chains of a planar lattice L 47 Cube K cubic extension of K D class (variety) of distributive lattices 24 Diag(K) diagonal embedding of K into Cube K Down P ordered set of down sets of the ordered set P 5, 9 ext: Con K Con L for K L, extension map: α conL(α) 42 fil(a)! filter generated≤ by the element7!a 15 fil(H) filter generated by the set H 15 FreeD(3) free distributive lattice on three generators 25 FreeK(H) free lattice generated by H in a variety K 27 FreeM(3) free modular lattice on three generators 27 Frucht C Frucht lattice of a graph C homf_;0g(X; Y ) the set of ; 0 -homomorphisms of X into Y id(a) ideal generatedf_ g by the element a 14 id(H) ideal generated by the set H 14 Id L ideal lattice of L 14 (Id) condition to define ideals 14 Isoform class of isoform lattices xiv Table of Notation Symbol Explanation Page J(D) ordered set of join-irreducible elements of D 19 J(γ) J(γ): J(E) J(D), the \inverse" of γ : D E 31 J(a) set of join-irreducible! elements below a ! 19 ker(γ) congruence kernel of γ 17 L class (variety) of all lattices 25 Lbottom;Ltop bottom and top of a rectangular lattice L 53 lc(L) left corner of a rectangular lattice L 53 Lleft;Lright left and right of a rectangular lattice L 53 M class (variety) of modular lattices 25 Max maximal elements of an ordered set mcr(n) minimal congruence representation function mcr(n; V) mcr for a class V M(D) ordered set of meet-irreducible elements of D 32 M3 five-element modular nondistributive lattice xvii, 11, 30 M3[L] ordered set of boolean triples of L M3[L; a] interval of M3[L] M3[L; a; b] interval of M3[L] M3[a; b] ordered set of boolean triples of the interval [a; b] 3 M3[α] restriction of α to M3[L] 3 M3[α; a] restriction of α to M3[L; a] 3 M3[α; a; b] restriction of α to M3[L; a; b] xvii N5 five-element nonmodular lattice xvii, 11, 20, 30 N5;5 seven-element nonmodular lattice N6 = N(p; q) six-element nonmodular lattice xvii N6[L] 2=3-boolean triple construction N(A; B) a lattice construction O(f) Landau big O notation xxviii Part A partition lattice of A 7, 9 Pow X power set lattice of X 5 Pow+ X ordered set of nonempty subsets of X Princ L ordered set of principal congruences L 38 rc(L) right corner of a rectangular lattice 52 Prime(L) set of prime intervals of L 39 re: Con L Con K restriction map: α α K 41 SecComp! class of sectionally complemented7! e lattices SemiMod class of semimodular lattices Simp K simple extension of K (SP_) join-substitution property 15 (SP^) meet-substitution property 15 sub(H) sublattice generated by H 13 S7 seven-element semimodular lattice xvii, 34 S8 eight-element semimodular lattice 34 x Swing, x p q, p swings to q T class (variety) of trivial lattices Uniform class of uniform lattices Traj L set of all trajectories of L 48 Table of Notation xv Symbol Explanation Page Relations and Congruences A2 set of ordered pairs of A 3 %; τ; π; : : : binary relations α; β;::: congruences 0 zero of Part A and Con L 8 1 unit of Part A and Con L 8 a b (mod π) a and b in the same block of π 7 a %≡ b a and b in relation % 3 a b (mod α) a and b in relation α 3 a/π≡ block containing a 7, 15 H/π blocks represented by H 7 α β product of α and β 21 ◦r α β reflexive product of α and β 29 ◦ α K restriction of α to the sublattice K 15 L=eα quotient lattice 16 β=α quotient congruence 17 πi projection map: L1 Ln Li 20 α β direct product of congruences× · · · × ! 21 × Ordered sets ; < ordering 3 ≤; > ordering, inverse notation 3 K≥ LK a sublattice of L 13 ≤ Q ordering of P restricted to a subset Q 4 a≤ b a incomparable with b 3 a k b a is covered by b 5 b ≺ a b covers a 5 0 zero, least element of an ordered set 4 1 unit, largest element of an ordered set 4 a b join operation 9 W_H least upper bound of H 3 a b meet operation 9 V^H greatest lower bound of H 4 P d dual of the ordered set (lattice) P 4, 10 [a; b] interval 14 H down set generated by H 5 #a down set generated by a 5 # f g P ∼= Q ordered set (lattice) P isomorphic to Q 4, 12 xvi Table of Notation Symbol Explanation Page Constructions P Q direct product of P and Q 5, 20 P +× Q sum of P and Q 6 P u Q glued sum of P and Q 17 A[B] tensor extension of A by B A B tensor product of A and B ⊗ U ~ V modular lattice construction Prime intervals p; q;::: prime intervals con(p) principal congruence generated by p 39 Prime(L) set of prime intervals of L 39 Princ L the ordered set of principal congruences 38 Perspectivities [a; b] s [c; d][a; b] perspective to [c; d] 32 up [a; b] s [c; d][a; b] up-perspective to [c; d] 32 dn [a; b] s [c; d][a; b] down-perspective to [c; d] 32 [a; b] t [c; d][a; b] projective to [c; d] 32 [a; b] [c; d][a; b] congruence-perspective onto [c; d] 36 up [a; b] [c; d][a; b] up congruence-perspective onto [c; d] 35 dn [a; b] [c; d][a; b] down congruence-perspective onto [c; d] 35 [a; b] [c; d][a; b] congruence-projective onto [c; d] 35 p ) p q p prime-perspective to q −!p-up p q p prime-perspective up to q −!p-dn p q p prime-perspective down to q −!p p = q p prime-projective to q p )x q p swings to q in x p q internal swing exx p q external swing Miscellaneous x closure of x 12 ? empty set 4 Picture Gallery C2 B2 C3 N5 N5,5 N6 S7 S8 M3 Preface to the Second Edition A few years after the publication of the first edition of this book, I submitted a paper on congruence lattices of finite lattices to a journal. The referee expressed surprise that I have more to say on this topic, after all, I published a whole book on the subject. This Second Edition, will attest to the fact the we, indeed, have a lot more to say on this subject.
Load more

Recommended publications
  • 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).
    [Show full text]
  • Semimodular Lattices Theory and Applications
    P1: SDY/SIL P2: SDY/SJS QC: SSH/SDY CB178/Stern CB178-FM March 3, 1999 9:46 ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Semimodular Lattices Theory and Applications MANFRED STERN P1: SDY/SIL P2: SDY/SJS QC: SSH/SDY CB178/Stern CB178-FM March 3, 1999 9:46 PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia c Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United States of America Typeset in 10/13 Times Roman in LATEX2ε[TB] A catalog record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Stern, Manfred. Semimodular lattices: theory and applications. / Manfred Stern. p. cm. – (Encyclopedia of mathematics and its applications ; v. 73) Includes bibliographical references and index. I. Title. II. Series. QA171.5.S743 1999 511.303 – dc21 98-44873 CIP ISBN 0 521 46105 7 hardback P1: SDY/SIL P2: SDY/SJS QC: SSH/SDY CB178/Stern CB178-FM March 3, 1999 9:46 Contents Preface page ix 1 From Boolean Algebras to Semimodular Lattices 1 1.1 Sources of Semimodularity
    [Show full text]
  • Chain Based Lattices
    Pacific Journal of Mathematics CHAIN BASED LATTICES GEORGE EPSTEIN AND ALFRED HORN Vol. 55, No. 1 September 1974 PACIFIC JOURNAL OF MATHEMATICS Vol. 55, No. 1, 1974 CHAIN BASED LATTICES G. EPSTEIN AND A. HORN In recent years several weakenings of Post algebras have been studied. Among these have been P0"lattices by T. Traezyk, Stone lattice of order n by T. Katrinak and A. Mitschke, and P-algebras by the present authors. Each of these system is an abstraction from certain aspects of Post algebras, and no two of them are comparable. In the present paper, the theory of P0-lattices will be developed further and two new systems, called Pi-lattices and P2-lattices are introduced. These systems are referred to as chain based lattices. P2-lattices form the intersection of all three weakenings mentioned above. While P-algebras and weaker systems such as L-algebras, Heyting algebras, and P-algebras, do not require any distinguished chain of elements other than 0, 1, chain based lattices require such a chain. Definitions are given in § 1. A P0-lattice is a bounded distributive lattice A which is generated by its center and a finite subchain con- taining 0 and 1. Such a subchain is called a chain base for A. The order of a P0-lattice A is the smallest number of elements in a chain base of A. In § 2, properties of P0-lattices are given which are used in later sections. If a P0-lattice A is a Heyting algebra, then it is shown in § 3, that there exists a unique chain base 0 = e0 < ex < < en_x — 1 such that ei+ί —* et = et for all i > 0.
    [Show full text]
  • Groups with Almost Modular Subgroup Lattice Provided by Elsevier - Publisher Connector
    Journal of Algebra 243, 738᎐764Ž. 2001 doi:10.1006rjabr.2001.8886, available online at http:rrwww.idealibrary.com on View metadata, citation and similar papers at core.ac.uk brought to you by CORE Groups with Almost Modular Subgroup Lattice provided by Elsevier - Publisher Connector Francesco de Giovanni and Carmela Musella Dipartimento di Matematica e Applicazioni, Uni¨ersita` di Napoli ‘‘Federico II’’, Complesso Uni¨ersitario Monte S. Angelo, Via Cintia, I 80126, Naples, Italy and Yaroslav P. Sysak1 Institute of Mathematics, Ukrainian National Academy of Sciences, ¨ul. Tereshchenki¨ska 3, 01601 Kie¨, Ukraine Communicated by Gernot Stroth Received November 14, 2000 DEDICATED TO BERNHARD AMBERG ON THE OCCASION OF HIS 60TH BIRTHDAY 1. INTRODUCTION A subgroup of a group G is called modular if it is a modular element of the lattice ᑦŽ.G of all subgroups of G. It is clear that everynormal subgroup of a group is modular, but arbitrarymodular subgroups need not be normal; thus modularitymaybe considered as a lattice generalization of normality. Lattices with modular elements are also called modular. Abelian groups and the so-called Tarski groupsŽ i.e., infinite groups all of whose proper nontrivial subgroups have prime order. are obvious examples of groups whose subgroup lattices are modular. The structure of groups with modular subgroup lattice has been described completelybyIwasawa wx4, 5 and Schmidt wx 13 . For a detailed account of results concerning modular subgroups of groups, we refer the reader towx 14 . 1 This work was done while the third author was visiting the Department of Mathematics of the Universityof Napoli ‘‘Federico II.’’ He thanks the G.N.S.A.G.A.
    [Show full text]
  • Meet Representations in Upper Continuous Modular Lattices James Elton Delany Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1966 Meet representations in upper continuous modular lattices James Elton Delany Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Delany, James Elton, "Meet representations in upper continuous modular lattices " (1966). Retrospective Theses and Dissertations. 2894. https://lib.dr.iastate.edu/rtd/2894 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been microiihned exactly as received 66-10,417 DE LA NY, James Elton, 1941— MEET REPRESENTATIONS IN UPPER CONTINU­ OUS MODULAR LATTICES. Iowa State University of Science and Technology Ph.D., 1966 Mathematics University Microfilms, Inc.. Ann Arbor, Michigan MEET REPRESENTATIONS IN UPPER CONTINUOUS MODULAR LATTICES by James Elton Delany A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Mathematics Approved: Signature was redacted for privacy. I _ , r Work Signature was redacted for privacy. Signature was redacted for privacy. Iowa State University of Science and Technology Ames, Iowa 1966 il TABLE OP CONTENTS Page 1. INTRODUCTION 1 2. TORSION ELEMENTS, TORSION FREE ELEMENTS, AND COVERING CONDITIONS 4 3. ATOMIC LATTICES AND UPPER CONTINUITY 8 4. COMPLETE JOIN HOMOMORPHISMS 16 5. TWO HOMOMORPHISMS 27 6.
    [Show full text]
  • This Is the Final Preprint Version of a Paper Which Appeared at Algebraic & Geometric Topology 17 (2017) 439-486
    This is the final preprint version of a paper which appeared at Algebraic & Geometric Topology 17 (2017) 439-486. The published version is accessible to subscribers at http://dx.doi.org/10.2140/agt.2017.17.439 SIMPLICIAL COMPLEXES WITH LATTICE STRUCTURES GEORGE M. BERGMAN Abstract. If L is a finite lattice, we show that there is a natural topological lattice structure on the geo- metric realization of its order complex ∆(L) (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of L: We note properties of this construction and of some variants, and pose several questions. For M3 the 5-element nondistributive modular lattice, ∆(M3) is modular, but its underlying topological space does not admit a structure of distributive lattice, answering a question of Walter Taylor. We also describe a construction of \stitching together" a family of lattices along a common chain, and note how ∆(M3) can be regarded as an example of this construction. 1. A lattice structure on ∆(L) I came upon the construction studied here from a direction unrelated to the concept of order complex; so I will first motivate it in roughly the way I discovered it, then recall the order complex construction, which turns out to describe the topological structures of these lattices. 1.1. The construction. The motivation for this work comes from Walter Taylor's paper [20], which ex- amines questions of which topological spaces { in particular, which finite-dimensional simplicial complexes { admit various sorts of algebraic structure, including structures of lattice. An earlier version of that paper asked whether there exist spaces which admit structures of lattice, but not of distributive lattice.
    [Show full text]
  • 2-Semilattices: Residual Properties and Applications to the Constraint Satisfaction Problem
    2-Semilattices: Residual Properties and Applications to the Constraint Satisfaction Problem by Ian Payne A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Pure Mathematics Waterloo, Ontario, Canada, 2017 c Ian Payne 2017 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner George McNulty Professor University of South Carolina Supervisor Ross Willard Professor University of Waterloo Internal Member Stanley Burris Distinguished Professor Emeritus Adjunct Professor University of Waterloo Internal Member Rahim Moosa Professor University of Waterloo Internal-external Member Chris Godsil Professor University of Waterloo ii I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract Semilattices are algebras known to have an important connection to partially ordered sets. In particular, if a partially ordered set (A; ≤) has greatest lower bounds, a semilattice (A; ^) can be associated to the order where a ^ b is the greatest lower bound of a and b. In this thesis, we study a class of algebras known as 2-semilattices, which is a generalization of the class of semilattices. Similar to the correspondence between partial orders and semilattices, there is a correspondence between certain digraphs and 2-semilattices. That is, to every 2-semilattice, there is an associated digraph which holds information about the 2-semilattice.
    [Show full text]
  • The Variety of Modular Lattices Is Not Generated
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 255, November 1979 THE VARIETYOF MODULAR LATTICES IS NOT GENERATED BY ITS FINITE MEMBERS BY RALPH FREESE1 Abstract. This paper proves the result of the title. It shows that there is a five-variable lattice identity which holds in all finite modular lattices but not in all modular lattices. It is also shown that every free distributive lattice can be embedded into a free modular lattice. An example showing that modular lattice epimorphisms need not be onto is given. We prove the result of the title by constructing a simple modular lattice of length six not in the variety generated by all finite modular lattices. This lattice can be generated by five elements and thus the free modular lattice on five generators, FM (5), is not residually finite. Our lattice is constructed using the technique of Hall and Dilworth [9] and is closely related to their third example. Let F and K be countably infinite fields of characteristics p and q, where p and q are distinct primes. Let Lp be the lattice of subspaces of a four-dimensional vector space over F, Lq the lattice of subspaces of a four-dimensional vector space over K. Two-dimen- sional quotients (i.e. intervals) in both lattices are always isomorphic to Mu (the two-dimensional lattice with « atoms). Thus Lp and Lq may be glued together over a two-dimensional quotient via [9], and this is our lattice. Notice that if F and K were finite fields we could not carry out the above construction since two-dimensional quotients of Lp would have/)" + 1 atoms and those of Lq would have qm + 1, for some n, m > 1.
    [Show full text]
  • Lattices: an Introduction
    Lattices: An Introduction Felix Gotti [email protected] UC Berkeley Student Combinatorics Seminar Felix Gotti [email protected] Lattices: An Introduction Outline General Lattices Modular Lattices Distributive Lattices Felix Gotti [email protected] Lattices: An Introduction Joins and Meets Definition (Joins and Meets) Let P be a poset. I If S ⊆ P a join (or supremum) of S, denoted by _ s; s2S is an element of u 2 P that is an upper bound of S satisfying that if u0 is any other upper bound of S, then u ≤ u0. I The definition of a meet (or infimum) of S ⊆ P, denoted by ^ s; s2S is dual to the definition of join. Remark: Note that if a join (resp., meet) exists then it is unique. Felix Gotti [email protected] Lattices: An Introduction Definition of Lattice Definition I A join-semilattice (resp., meet-semilattice) is a poset such that any pair of elements have a join (resp., meet). I A lattice is a poset that is both a join-semilattice and a meet-semilattice. I If L is a lattice and S ⊂ L such that r _ s; r ^ s 2 S for all r; s 2 S, we say that S is a sublattice of L. Example of lattices: I Every totally ordered set is a lattice. ∗ I If L and M are lattices, so are L ; L ⊕ M, and L × M. While L + M is not a lattice, at least L or M is empty, (L + M) [ f0^; 1^g is always a lattice. I The lattices L and M are sublattices of L ⊕ M and (L + M) [ f0^; 1^g.
    [Show full text]
  • DISTRIBUTIVE LATTICES FACT 1: for Any Lattice <A,≤>: 1 and 2 and 3 and 4 Hold in <A,≤>: the Distributive Inequal
    DISTRIBUTIVE LATTICES FACT 1: For any lattice <A,≤>: 1 and 2 and 3 and 4 hold in <A,≤>: The distributive inequalities: 1. for every a,b,c ∈ A: (a ∧ b) ∨ (a ∧ c) ≤ a ∧ (b ∨ c) 2. for every a,b,c ∈ A: a ∨ (b ∧ c) ≤ (a ∨ b) ∧ (a ∨ c) 3. for every a,b,c ∈ A: (a ∧ b) ∨ (b ∧ c) ∨ (a ∧ c) ≤ (a ∨ b) ∧ (b ∨ c) ∧ (a ∨ c) The modular inequality: 4. for every a,b,c ∈ A: (a ∧ b) ∨ (a ∧ c) ≤ a ∧ (b ∨ (a ∧ c)) FACT 2: For any lattice <A,≤>: 5 holds in <A,≤> iff 6 holds in <A,≤> iff 7 holds in <A,≤>: 5. for every a,b,c ∈ A: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) 6. for every a,b,c ∈ A: a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) 7. for every a,b,c ∈ A: a ∨ (b ∧ c) ≤ b ∧ (a ∨ c). A lattice <A,≤> is distributive iff 5. holds. FACT 3: For any lattice <A,≤>: 8 holds in <A,≤> iff 9 holds in <A,≤>: 8. for every a,b,c ∈ A:(a ∧ b) ∨ (a ∧ c) = a ∧ (b ∨ (a ∧ c)) 9. for every a,b,c ∈ A: if a ≤ b then a ∨ (b ∧ c) = b ∧ (a ∨ c) A lattice <A,≤> is modular iff 8. holds. FACT 4: Every distributive lattice is modular. Namely, let <A,≤> be distributive and let a,b,c ∈ A and let a ≤ b. a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) [by distributivity] = b ∧ (a ∨ c) [since a ∨ b =b]. The pentagon: The diamond: o 1 o 1 o z o y x o o y o z o x o 0 o 0 THEOREM 5: A lattice is modular iff the pentagon cannot be embedded in it.
    [Show full text]
  • Modular Lattices
    Wednesday 1/30/08 Modular Lattices Definition: A lattice L is modular if for every x; y; z 2 L with x ≤ z, (1) x _ (y ^ z) = (x _ y) ^ z: (Note: For all lattices, if x ≤ z, then x _ (y ^ z) ≤ (x _ y) ^ z.) Some basic facts and examples: 1. Every sublattice of a modular lattice is modular. Also, if L is distributive and x ≤ z 2 L, then x _ (y ^ z) = (x ^ z) _ (y ^ z) = (x _ y) ^ z; so L is modular. 2. L is modular if and only if L∗ is modular. Unlike the corresponding statement for distributivity, this is completely trivial, because the definition of modularity is invariant under dualization. 3. N5 is not modular. With the labeling below, we have a ≤ b, but a _ (c ^ b) = a _ 0^ = a; (a _ c) ^ b = 1^ ^ b = b: b c a ∼ 4. M5 = Π3 is modular. However, Π4 is not modular (exercise). Modular lattices tend to come up in algebraic settings: • Subspaces of a vector space • Subgroups of a group • R-submodules of an R-module E.g., if X; Y; Z are subspaces of a vector space V with X ⊆ Z, then the modularity condition says that X + (Y \ Z) = (X + Y ) \ Z: Proposition 1. Let L be a lattice. TFAE: 1. L is modular. 2. For all x; y; z 2 L, if x 2 [y ^ z; z], then x = (x _ y) ^ z. 2∗. For all x; y; z 2 L, if x 2 [y; y _ z], then x = (x ^ z) _ y.
    [Show full text]
  • Algebraic Lattices in QFT Renormalization
    Algebraic lattices in QFT renormalization Michael Borinsky∗ Institute of Physics, Humboldt University Newton Str. 15, D-12489 Berlin, Germany Abstract The structure of overlapping subdivergences, which appear in the perturbative expan- sions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the Standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams. Keywords. quantum field theory, renormalization, Hopf algebra of Feynman diagrams, algebraic lattices, zero-dimensional QFT MSC. 81T18, 81T15, 81T16, 06B99 1 Introduction Calculations of observable quantities in quantum field theory rely almost always on perturbation theory. The integrals in the perturbation expansions can be depicted as Feynman diagrams. Usually these integrals require renormalization to give meaningful results. The renormalization procedure can be performed using the Hopf algebra of Feynman graphs [10], which organizes the classic BPHZ procedure into an algebraic framework. The motivation for this paper was to obtain insights on the coradical filtration of this Hopf algebra and thereby on the structure of the subdivergences of the Feynman diagrams in the perturbation expansion. The perturbative expansions in QFT are divergent series themselves as first pointed out by [12].
    [Show full text]