A Primer of Subquasivariety Lattices (CMS/CAIMS Books in Mathematics, 3)

This book addresses Birkhoff and Mal'cev's problem of describing subquasivariety lattices. The text begins by developing the basics of atomic theories and implicational theories in languages that may, or may not, contain equality. Subquasivariety lattices are represented as lattices of closed algebraic subsets of a lattice with operators, which yields new restrictions on the equaclosure operator. As an application of this new approach, it is shown that completely distributive lattices with a dually compact least element are subquasivariety lattices. The book contains many examples to illustrate these principles, as well as open problems. Ultimately this new approach gives readers a set of tools to investigate classes of lattices that can be represented as subquasivariety lattices. Preface 6 Contents 7 1 Introduction 10 1.1 An Overview 11 1.2 Lattices, Theories, and Models 12 1.3 A Review of Classical Subvariety Lattices 13 1.4 A Review of Classical Subquasivariety Lattices 15 1.5 It's Complicated: The Complexity of Subquasivariety Lattices 19 1.6 A Review of Lattices of Algebraic Sets 26 1.7 Fully Invariant Elements 32 1.8 Finite Lower Bounded Lattices 33 2 Varieties and Quasivarieties in General Languages 35 2.1 Basic Universal Algebra 35 2.1.1 Substructures and Direct Products 37 2.1.2 Congruence Lattices from the Beginning 38 2.1.3 Equimorphism 43 2.2 Freedom's Just Another Word 47 2.3 Theories 51 2.3.1 Atomic Theories 53 2.3.2 Implicational Theories 54 2.3.3 The Converse 58 2.4 Models 59 2.5 Two Quasivarieties Without Equality 64 2.5.1 Quasivarieties of Prequivalences 64 2.5.2 A Variety of Unary Structures 66 2.6 Basic Properties of Subquasivariety Lattices 69 2.7 Atomistic and Finite Distributive Subquasivariety Lattices 71 2.8 The Lattice Sp(S,H) Is Dually Algebraic 74 3 Equaclosure Operators 77 3.1 Natural Equaclosure Operators 78 3.2 The H-Closed Algebraic Subset Generated by a Set 84 3.3 Companion Lattices 88 3.4 A New Condition for Equaclosure Operators 91 3.5 Directed Meets in τ(L) 97 3.6 More Properties of Equaclosure Operators 97 3.6.1 Three Properties of Weak Equaclosure Operators 97 3.6.2 An Almost Old Observation 99 4 Preclops on Finite Lattices 100 4.1 Meet Distributive Elements and Preclops 101 4.1.1 Some Finite Join Semidistributive Lattices That Support No Preclop 107 4.1.2 Examples and Sufficient Conditions 109 4.1.3 Finite Atomistic Lattices 110 4.1.4 Necessary Conditions 112 4.2 Algorithm to Determine Whether a Lattice has a Preclop 113 4.2.1 0-Separating Homomorphisms 117 4.3 Embedding a Lower Bounded Lattice into Sub S 118 4.4 A General Embedding Method 126 4.4.1 A Semilattice Example 127 4.4.2 Group Examples 129 4.4.3 Isomorphism 130 5 Finite Lattices as Sub(S,,1,H): The Case J(L) τ(L) 132 5.1 Companion Lattices Rise Again 134 5.2 An Algorithm to Test for Representability When J(L) τ(L) 138 5.3 Extension to a Class of Infinite Lattices 142 6 Finite Lattices as Sub(S,,1,H): The Case J(L) τ(L) 146 6.1 Ad hoc Representations 146 6.2 Representations Based on Embeddings into SubS 152 6.2.1 (W,μ) 155 6.2.2 (W,γ2) 156 6.2.3 (W,γ3) 158 6.2.4 (W,γ4) 160 6.2.5 (W,γ5) 161 6.2.6 (W,γ6) 163 6.2.7 (W,γ1) 164 7 The Six-Step Program: From (L,γ) to (Lq(K),Γ) 167 7.1 The Six-Step Program 168 7.1.1 Step 1 168 7.1.2 Step 2 171 7.1.3 Step 3 173 7.1.4 Step 4 178 7.1.5 Step 5 179 7.1.6 Step 6 181 7.2 A Case Where Step Six Works: Longstyle 187 7.3 Reverse Engineering 195 7.4 A Case Where Step Six Works: Shortstyle 198 7.5 Shortstyle Representations Revisited 207 7.6 Mediumstyle Representations 215 8 Lattices 1 + L as Lq(K) 223 8.1 The Leaf Lattice and Generalizations 224 8.2 Examples of Representations for 1 + L 228 8.2.1 The Lattice 1 + Co(2 2) 228 8.2.2 The Lattice 1 + J 228 8.2.3 The Lattice 1 + (2 3) 230 8.2.4 1 + O(P) 231 8.2.5 The Lattice 1 + H, the Hexagon 233 8.2.6 1 + L with L Subdirectly Irreducible, Rank 1 234 8.2.7 1 + L with L Subdirectly Reducible, Rank 1 239 8.3 An Age-Old Question Answered 241 9 Representing Distributive Dually Algebraic Lattices 243 9.1 Distributive Dually Algebraic Lattices as Sp(S,H) 243 9.2 Lattices of Order Ideals as Sp(S,H) 250 9.3 Lattices of Order Ideals as Lq(K) 251 9.4 Ideals of Meet Semilattices as Lq(K) 256 9.5 Recapitulation on Distributive, Dually Algebraic Lattices 257 10 Problems and an Advertisement 260 10.1 Problems 260 10.2 Advertisement 263 Appendices 265 A.1 Additional Examples 266 A.2 Lattices of Atomic Theories in Languages Without Equality 270 A.2.1 Atomic Theories 270 A.2.2 Lattices of Atomic Theories 271 A.2.3 Conclusion 274 A.3 In Search of Quasicriticality 274 A.3.1 Quasicriticality 274 A.3.2 Strong Quasicriticality 276 A.3.3 Three Examples 276 Bibliography 281 Symbol Index 286 Author Index 288 Subject Index 290