The Congruences of a Finite Lattice George Gr¨atzer Birkh¨auser Verlag, Boston @ 2005 Birkh¨auser Verlag To L´aszl´o Fuchs, my thesis advisor, my teacher, who taught me to set the bar high; and to the coauthors of my papers, Tomi (E. T. Schmidt), Harry (H. Lakser), Ervin (E. Fried), David (D. Kelly), Bob (R. W. Quackenbush), Fred (Wehrung), and more than fifty others, who helped me raise the bar. Contents Table of Notation xiii Picture Gallery xvii Acknowledgments xix Preface xxi I A Brief Introduction to Lattices 1 1 Basic Concepts 3 1.1 Ordering 3 1.1.1 Orders 3 1.1.2 Diagrams 5 1.1.3 Order constructions 5 1.1.4 Partitions 6 1.2 Lattices and semilattices 8 1.2.1 Lattices 8 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 14 2 Special Concepts 19 2.1 Elements and lattices 19 2.2 Direct and subdirect products 20 2.3 Polynomials and identities 23 2.4 Gluing 26 vii viii 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 Prime intervals 37 3.3 Congruence-preserving extensions and variants 39 II Basic Techniques 45 4 Chopped Lattices 47 4.1 Basic definitions 47 4.2 Compatible vectors of elements 49 4.3 Compatible vectors of congruences 50 4.4 From the chopped lattice to the ideal lattice 52 4.5 Sectional complementation 53 5 Boolean Triples 57 5.1 The general construction 57 5.2 The congruence-preserving extension property 60 5.3 The distributive case 62 5.4 Two interesting intervals 63 6 Cubic Extensions 71 6.1 The construction 71 6.2 The basic property 73 III Representation Theorems 77 7 The Dilworth Theorem 79 7.1 The representation theorem 79 7.2 Proof-by-Picture 80 7.3 Computing 82 7.4 Sectionally complemented lattices 83 7.5 Discussion 85 8 Minimal Representations 93 8.1 The results 93 8.2 Proof-by-Picture for minimal construction 94 8.3 The formal construction 95 8.4 Proof-by-Picture for minimality 97 Contents ix 8.5 Computing minimality 99 8.6 Discussion 100 9 Semimodular Lattices 105 9.1 The representation theorem 105 9.2 Proof-by-Picture 106 9.3 Construction and proof 107 9.4 Discussion 114 10 Modular Lattices 115 10.1 The representation theorem 115 10.2 Proof-by-Picture 116 10.3 Construction and proof 120 10.4 Discussion 125 11 Uniform Lattices 129 11.1 The representation theorem 129 11.2 Proof-by-Picture 129 11.3 The lattice N(A, B) 132 11.4 Formal proof 137 11.5 Discussion 139 IV Extensions 143 12 Sectionally Complemented Lattices 145 12.1 The extension theorem 145 12.2 Proof-by-Picture 146 12.3 Simple extensions 148 12.4 Formal proof 150 12.5 Discussion 152 13 Semimodular Lattices 153 13.1 The extension theorem 153 13.2 Proof-by-Picture 153 13.3 The conduit 156 13.4 The construction 157 13.5 Formal proof 159 13.6 Discussion 159 14 Isoform Lattices 161 14.1 The result 161 14.2 Proof-by-Picture 161 14.3 Formal construction 165 14.4 The congruences 171 x Contents 14.5 The isoform property 172 14.6 Discussion 173 15 Independence Theorems 177 15.1 Results 177 15.2 Proof-by-Picture 178 15.2.1 Frucht lattices 178 15.2.2 An automorphism-preserving simple extension 179 15.2.3 A congruence-preserving rigid extension 180 15.2.4 Merging the two extensions 181 15.2.5 The representation theorems 182 15.3 Formal proofs 183 15.3.1 An automorphism-preserving simple extension 183 15.3.2 A congruence-preserving rigid extension 185 15.3.3 Proof of the independence theorems 185 15.4 Discussion 187 16 Magic Wands 189 16.1 Constructing congruence lattices 189 16.1.1 Bijective maps 189 16.1.2 Surjective maps 190 16.2 Proof-by-Picture for bijective maps 191 16.3 Verification for bijective maps 194 16.4 2/3-boolean triples 198 16.5 Proof-by-Picture for surjective maps 204 16.6 Verification for surjective maps 206 16.7 Discussion 207 V Two Lattices 213 17 Sublattices 215 17.1 The results 215 17.2 Proof-by-Picture 217 17.3 Multi-coloring 219 17.4 Formal proof 220 17.5 Discussion 221 18 Ideals 227 18.1 The results 227 18.2 Proof-by-Picture for the main result 228 18.3 A very formal proof: Main result 230 18.3.1 Categoric preliminaries 230 18.3.2 From DI to OR 232 Contents xi 18.3.3 From OR to HE 232 18.3.4 From CH to DI 233 18.3.5 From HE to CH 234 18.3.6 From CH to LA 237 18.3.7 The final step 237 18.4 Proof for sectionally complemented lattices 238 18.5 Proof-by-Picture for planar lattices 241 18.6 Discussion 242 19 Tensor Extensions 245 19.1 The problem 245 19.2 Three unary functions 246 19.3 Defining tensor extensions 248 19.4 Computing 250 19.4.1 Some special elements 250 19.4.2 An embedding 252 19.4.3 Distributive lattices 253 19.5 Congruences 254 19.5.1 Congruence spreading 254 19.5.2 Some structural observations 257 19.5.3 Lifting congruences 259 19.5.4 The main lemma 261 19.6 The congruence isomorphism 262 19.7 Discussion 263 Bibliography 265 Index 275 Table of Notation Symbol Explanation Page Atom(U) set of atoms of the ideal U 84 Aut L automorphism group of L 12 Bn boolean lattice with n atoms 4 Cn n-element chain 4 con(a, b) smallest congruence under which a ≡ b 15 con(c) principal congruence for a color c 39 con(H) smallest congruence collapsing H 16 con(p) principal congruence for the prime interval p 37 Con L congruence lattice of L 15, 48 ConJ L order of join-irreducible congruences of L 37 ConM L order of meet-irreducible congruences of L 71 Cube K cubic extension of K 71 D class (variety) of distributive lattices 24 Diag diagonal embedding of K into Cube K 71 Down P order of down-sets of the (hemi)order P 4, 9, 232 ext: Con K → Con L for K ≤ L, extension map: Θ → conL(Θ) 41 fil(a) filter generated by the element a 14 fil(H) filter generated by the set H 14 FD(3) free distributive lattice on three generators 26 FK(H) free lattice generated by H in a variety K 26 FM(3) free modular lattice on three generators 28 Frucht C Frucht lattice of a graph C 178 hom{∨,0}(X, Y ) {∨, 0}-homomorphism of X into Y 253 xiii xiv Table of Notation Symbol Explanation Page id(a) ideal generated by the element a 14 id(H) ideal generated by the set H 14 Id L ideal lattice of L 14, 48 (Id) condition to define ideals 14, 48 Isoform class of isoform lattices 141 J(D) order of join-irreducible elements of D 19 J(ϕ)J(ϕ): J(E) → J(D), the “inverse” of ϕ: D → E 32 J(a) set of join-irreducible elements below a 19 ker(ϕ) congruence kernel of ϕ 16 L class (variety) of all lattices 25 M class (variety) of modular lattices 25 Max maximal elements of an order 49 mcr(n) minimal congruence representation function 87 mcr(n, V) mcr for a class V 87 M(D) order of meet-irreducible elements of D 32 M3 five-element modular nondistributive lattice xvii, 11, 30 M3[L] order of boolean triples of L 58 M3[L, a] interval of M3[L]63 M3[L, a, b] interval of M3[L]65 M3[a, b] order of boolean triples of the interval [a, b]58 3 M3[Θ] reflection of Θ to M3[L]60 3 M3[Θ,a] reflection of Θ to M3[L, a]64 3 M3[Θ,a,b] reflection of Θ to M3[L, a, b] xvii, 67 N5 five-element nonmodular lattice xvii, 11, 30 N5,5 seven-element nonmodular lattice 94 N6 = N(p, q) six-element nonmodular lattice xvii, 80 N6[L]2/3-boolean triple construction 198 N(A, B) lattice construction 132 O(f) Landau O notation xxvi Part A partition lattice of A 7, 9 Pow X power set lattice of X 4 Pow+ X order of nonempty subsets of X 219 Prime(L) set of prime intervals of L 37 re: Con L → Con K reflection (restriction) map: Θ → ΘK 39 SecComp class of sectionally complemented lattices 87 SemiMod class of semimodular lattices 87 Simp K simple extension of K 71 (SP∨) join-substitution property 14, 48 (SP∧) meet-substitution property xvii, 14, 48 sub(H) sublattice generated by H 13 S8 eight-element semimodular lattice 106 T class (variety) of trivial lattices 25 Uniform class of uniform lattices 141 Table of Notation xv Symbol Explanation Page Relations and Congruences A2 set of ordered pairs of A 3 , τ, π, . binary relations Θ, Ψ, . congruences ω zero of Part A 7 ι unit of Part A 7 a ≡ b (π) a and b in the same block of π 7 ab aand b in relation 3 a ≡ b (Θ) a and b in relation Θ 3 a/π block containing a 6, 14 H/π blocks represented by H 7 α ◦ β product of α and β 21 r α ◦ β reflexive product of α and β 30 ΘK restriction of Θ to the sublattice K 14 L/Θ quotient lattice 16 Φ/Θ quotient congruence 16 πi projection map: L1 ×···×Ln → Li 21 Θ × Φ direct product of congruences 21 Orders ≤, < ordering 3 ≥, > ordering, inverse notation 3 K ≤ LKa sublattice of L 13 ≤Q ordering of P restricted to a subset Q 4 a baincomparable with b 3 a ≺ bais covered by b 5 b abcovers a 5 0 zero, least element of an order 4 1 unit, largest element of an order 4 a ∨ b join operation 9 H least upper bound of H 3 a ∧ b meet operation 9 H greatest lower bound of H 4 P d dual of the order (lattice) P 4, 10 [a, b] interval 13 ↓H down-set generated by H 4 ↓a down-set generated by {a} 4 P =∼ Q order (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 Q glued sum of P and Q 16 A[B] tensor extension of A by B 248 A ⊗ B tensor product of A and B 245 U V modular lattice construction 120 Perpectivities [a, b] ∼ [c, d][a, b] perspective to [c, d]32 [a, b] ∼u [c, d][a, b] up-perspective to [c, d]33 [a, b] ∼d [c, d][a, b] down-perspective to [c, d]33 [a, b] ≈ [c, d][a, b] projective to [c, d]33 [a, b] [c, d][a, b] up congruence-perspective onto [c, d]35 [a, b] [c, d][a, b] down congruence-perspective onto [c, d]35 [a, b] → [c, d][a, b] congruence-perspective onto [c, d]35 [a, b] ⇒ [c, d][a, b] congruence-projective onto [c, d]36 [a, b] ⇔ [c, d][a, b] ⇒ [c, d] and [c, d] ⇒ [a, b]36 Prime intervals p, q, ..