A Primer of Subquasivariety Lattices

Description

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 Contents 1 Introduction 1.1 An Overview 1.2 Lattices, Theories, and Models 1.3 A Review of Classical Subvariety Lattices 1.4 A Review of Classical Subquasivariety Lattices 1.5 It's Complicated: The Complexity of Subquasivariety Lattices 1.6 A Review of Lattices of Algebraic Sets 1.7 Fully Invariant Elements 1.8 Finite Lower Bounded Lattices 2 Varieties and Quasivarieties in General Languages 2.1 Basic Universal Algebra 2.1.1 Substructures and Direct Products 2.1.2 Congruence Lattices from the Beginning 2.1.3 Equimorphism 2.2 Freedom's Just Another Word 2.3 Theories 2.3.1 Atomic Theories 2.3.2 Implicational Theories 2.3.3 The Converse 2.4 Models 2.5 Two Quasivarieties Without Equality 2.5.1 Quasivarieties of Prequivalences 2.5.2 A Variety of Unary Structures 2.6 Basic Properties of Subquasivariety Lattices 2.7 Atomistic and Finite Distributive Subquasivariety Lattices 2.8 The Lattice Sp(S,H) Is Dually Algebraic 3 Equaclosure Operators 3.1 Natural Equaclosure Operators 3.2 The H-Closed Algebraic Subset Generated by a Set 3.3 Companion Lattices 3.4 A New Condition for Equaclosure Operators 3.5 Directed Meets in τ(L) 3.6 More Properties of Equaclosure Operators 3.6.1 Three Properties of Weak Equaclosure Operators 3.6.2 An Almost Old Observation 4 Preclops on Finite Lattices 4.1 Meet Distributive Elements and Preclops 4.1.1 Some Finite Join Semidistributive Lattices That Support No Preclop 4.1.2 Examples and Sufficient Conditions 4.1.3 Finite Atomistic Lattices 4.1.4 Necessary Conditions 4.2 Algorithm to Determine Whether a Lattice has a Preclop 4.2.1 0-Separating Homomorphisms 4.3 Embedding a Lower Bounded Lattice into Sub S 4.4 A General Embedding Method 4.4.1 A Semilattice Example 4.4.2 Group Examples 4.4.3 Isomorphism 5 Finite Lattices as Sub(S,,1,H): The Case J(L) τ(L) 5.1 Companion Lattices Rise Again 5.2 An Algorithm to Test for Representability When J(L) τ(L) 5.3 Extension to a Class of Infinite Lattices 6 Finite Lattices as Sub(S,,1,H): The Case J(L) τ(L) 6.1 Ad hoc Representations 6.2 Representations Based on Embeddings into SubS 6.2.1 (W,μ) 6.2.2 (W,γ2) 6.2.3 (W,γ3) 6.2.4 (W,γ4) 6.2.5 (W,γ5) 6.2.6 (W,γ6) 6.2.7 (W,γ1) 7 The Six-Step Program: From (L,γ) to (Lq(K),Γ) 7.1 The Six-Step Program 7.1.1 Step 1 7.1.2 Step 2 7.1.3 Step 3 7.1.4 Step 4 7.1.5 Step 5 7.1.6 Step 6 7.2 A Case Where Step Six Works: Longstyle 7.3 Reverse Engineering 7.4 A Case Where Step Six Works: Shortstyle 7.5 Shortstyle Representations Revisited 7.6 Mediumstyle Representations 8 Lattices 1 + L as Lq(K) 8.1 The Leaf Lattice and Generalizations 8.2 Examples of Representations for 1 + L 8.2.1 The Lattice 1 + Co(2 2) 8.2.2 The Lattice 1 + J 8.2.3 The Lattice 1 + (2 3) 8.2.4 1 + O(P) 8.2.5 The Lattice 1 + H, the Hexagon 8.2.6 1 + L with L Subdirectly Irreducible, Rank 1 8.2.7 1 + L with L Subdirectly Reducible, Rank 1 8.3 An Age-Old Question Answered 9 Representing Distributive Dually Algebraic Lattices 9.1 Distributive Dually Algebraic Lattices as Sp(S,H) 9.2 Lattices of Order Ideals as Sp(S,H) 9.3 Lattices of Order Ideals as Lq(K) 9.4 Ideals of Meet Semilattices as Lq(K) 9.5 Recapitulation on Distributive, Dually Algebraic Lattices 10 Problems and an Advertisement 10.1 Problems 10.2 Advertisement Appendices A.1 Additional Examples A.2 Lattices of Atomic Theories in Languages Without Equality A.2.1 Atomic Theories A.2.2 Lattices of Atomic Theories A.2.3 Conclusion A.3 In Search of Quasicriticality A.3.1 Quasicriticality A.3.2 Strong Quasicriticality A.3.3 Three Examples Bibliography Symbol Index Author Index Subject Index