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

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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.
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