Practical Foundations of Mathematics

Paul Taylor

III. Posets and Lattices

INTRODUCTION

3.1    POSETS AND MONOTONE FUNCTIONS

• Monotone functions
• Representation of orders by subset-inclusion

3.2    MEETS, JOINS AND LATTICES

• Meets and joins
• Meets and joins of lower sets
• Diagrams
• Lattices

3.3    FIXED POINTS AND PARTIAL FUNCTIONS

• The poset of partial functions
• The fixed point theorem

3.4    DOMAINS

• Directed diagrams
• Posets with directed joins
• The Scott topology

3.5    PRODUCTS AND FUNCTION-SPACES

• Products of posets, domains, lattices and diagrams
• Pointwise meets and joins
• Joint continuity
• Scott's thesis

3.6    ADJUNCTIONS

• Order, composition and equivalence
• The adjoint function theorem
• Frames and Heyting algebras

3.7    CLOSURE CONDITIONS AND INDUCTION

• Closure conditions
• Induction on closures

3.8    MODALITIES AND GALOIS CONNECTIONS

• Modal logic
• The transitive closure
• Modal logic for preorders
• Functions and quantifiers
• Galois connections

3.9    CONSTRUCTIONS WITH CLOSURE CONDITIONS

• The Lindenbaum algebra for conjunction
• Algebraic lattices
• Adding and preserving joins
• Joins which are stable under meet
• Generalising from propositions to types

EXERCISES  III