Key Concepts of Complete Lattices

Complete lattices are special partially ordered sets where every subset has a supremum and an infimum. This concept extends traditional lattices, ensuring that even infinite subsets maintain these properties, making them essential in various mathematical fields.

1. Definition of a complete lattice

   • A complete lattice is a partially ordered set (poset) in which every subset has both a supremum (least upper bound) and an infimum (greatest lower bound).
   • It generalizes the concept of a lattice by ensuring completeness for all subsets, not just finite ones.
   • Notation: A complete lattice is often denoted as (L, ≤) where L is the set and ≤ is the order relation.

2. Supremum and infimum in complete lattices

   • The supremum (join) of a subset S is the least element in L that is greater than or equal to every element in S.
   • The infimum (meet) of a subset S is the greatest element in L that is less than or equal to every element in S.
   • In a complete lattice, every subset, including infinite ones, has both a supremum and an infimum.

3. Completeness property

   • The completeness property ensures that every subset of a complete lattice has a supremum and an infimum, making it robust for various mathematical applications.
   • This property is crucial for the study of fixed points, order theory, and topology.
   • Completeness distinguishes complete lattices from general lattices, which may not have this property.

4. Hasse diagrams for complete lattices

   • Hasse diagrams visually represent the elements of a lattice and their order relations, omitting transitive edges for clarity.
   • In complete lattices, Hasse diagrams can become complex due to the presence of infinite subsets.
   • They help in understanding the structure and relationships within the lattice.

5. Examples of complete lattices (e.g., power set lattice)

   • The power set of any set, ordered by inclusion, is a classic example of a complete lattice.
   • The set of all subsets of a given set has both a supremum (union) and an infimum (intersection) for any collection of subsets.
   • Other examples include the set of all real numbers with respect to the usual order and the set of all closed intervals in the real line.

6. Knaster-Tarski fixed-point theorem

   • This theorem states that any monotone function on a complete lattice has at least one fixed point.
   • It is significant in various fields, including computer science, economics, and game theory.
   • The theorem provides a powerful tool for proving the existence of solutions to certain equations and optimization problems.

7. Complete sublattices

   • A complete sublattice is a subset of a complete lattice that is itself a complete lattice.
   • It retains the properties of having suprema and infima for all its subsets.
   • Understanding complete sublattices helps in analyzing the structure and behavior of larger complete lattices.

8. Galois connections in complete lattices

   • A Galois connection is a pair of monotone functions between two complete lattices that establish a correspondence between their elements.
   • It provides a framework for understanding dualities and relationships between different mathematical structures.
   • Galois connections are useful in various applications, including fixed-point theory and algebra.

9. Distributive complete lattices

   • A distributive complete lattice is one where the join and meet operations distribute over each other.
   • This property simplifies many operations and proofs within the lattice.
   • Examples include the lattice of open sets in a topological space and the lattice of ideals in a ring.

10. Completion of a lattice

    • The completion of a lattice involves adding elements to ensure that every subset has a supremum and infimum.
    • This process can lead to the formation of a complete lattice from an incomplete one.
    • Techniques for completion include taking the set of all lower bounds or upper bounds of subsets, leading to a new structure that retains the original lattice's properties.