📊Order Theory Unit 3 – Lattices and their properties

What Are Lattices?

• Lattices are partially ordered sets where every pair of elements has a unique least upper bound (join) and greatest lower bound (meet)
• Consist of a set of elements and a binary relation that defines the partial order
• The partial order relation is reflexive, antisymmetric, and transitive
• Lattices can be represented visually using Hasse diagrams
  • Hasse diagrams depict the elements as nodes and the partial order relation as edges
  • If $a \leq b$, then $a$ appears below $b$ in the diagram
• Lattices have a top element (greatest element) and a bottom element (least element)
• Examples of lattices include:
  • The power set of a set under inclusion ($\subseteq$)
  • The set of natural numbers under divisibility ($\mid$)

Key Properties of Lattices

• Idempotence: For any element $a$ in a lattice, $a \vee a = a$ and $a \wedge a = a$
• Commutativity: For any elements $a$ and $b$ in a lattice, $a \vee b = b \vee a$ and $a \wedge b = b \wedge a$
• Associativity: For any elements $a$, $b$, and $c$ in a lattice, $(a \vee b) \vee c = a \vee (b \vee c)$ and $(a \wedge b) \wedge c = a \wedge (b \wedge c)$
• Absorption: For any elements $a$ and $b$ in a lattice, $a \vee (a \wedge b) = a$ and $a \wedge (a \vee b) = a$
• Boundedness: Every lattice has a top element $\top$ and a bottom element $\bot$
  • For any element $a$ in the lattice, $a \vee \top = \top$ and $a \wedge \bot = \bot$
• Distributivity: A lattice is distributive if for any elements $a$, $b$, and $c$, $a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$ and $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$
• Modular inequality: A lattice is modular if for any elements $a$, $b$, and $c$ with $a \leq c$, $(a \vee b) \wedge c = a \vee (b \wedge c)$

Types of Lattices

• Complete lattices: Every subset of elements has a join and a meet
  • Examples include the power set lattice and the real interval lattice $[0, 1]$
• Bounded lattices: Lattices with a top element and a bottom element
• Distributive lattices: Lattices that satisfy the distributive property
  • Examples include the lattice of subsets of a set and the lattice of divisors of a number
• Modular lattices: Lattices that satisfy the modular inequality
  • Distributive lattices are always modular, but the converse is not true
• Complemented lattices: Lattices where every element has a complement
  • In a complemented lattice, for any element $a$, there exists an element $b$ such that $a \vee b = \top$ and $a \wedge b = \bot$
• Boolean algebras: Complemented distributive lattices
  • Examples include the power set lattice and the lattice of propositions in logic
• Finite lattices: Lattices with a finite number of elements
• Infinite lattices: Lattices with an infinite number of elements

Lattice Operations and Diagrams

• Join operation ($\vee$): The least upper bound of two elements in a lattice
  • For elements $a$ and $b$, $a \vee b$ is the smallest element that is greater than or equal to both $a$ and $b$
  • In a Hasse diagram, the join of two elements is the first common node above both elements
• Meet operation ($\wedge$): The greatest lower bound of two elements in a lattice
  • For elements $a$ and $b$, $a \wedge b$ is the largest element that is less than or equal to both $a$ and $b$
  • In a Hasse diagram, the meet of two elements is the first common node below both elements
• Hasse diagrams: Visual representations of lattices
  • Elements are represented as nodes, and the partial order relation is represented by edges
  • If $a \leq b$, then $a$ appears below $b$ in the diagram, and there is an edge connecting them
  • Transitive edges are omitted for clarity
• Drawing Hasse diagrams:
  • Start with the bottom element (if it exists) and work upwards
  • Connect elements with edges based on the partial order relation
  • Omit transitive edges to simplify the diagram
• Reading Hasse diagrams:
  • If there is a path from $a$ to $b$, then $a \leq b$
  • The join of two elements is the first common node above both elements
  • The meet of two elements is the first common node below both elements

Sublattices and Homomorphisms

• Sublattices: A subset of a lattice that is itself a lattice under the same join and meet operations
  • A subset $S$ of a lattice $L$ is a sublattice if for any $a, b \in S$, $a \vee b \in S$ and $a \wedge b \in S$
  • Examples include the lattice of even numbers as a sublattice of the lattice of natural numbers under divisibility
• Lattice homomorphisms: Structure-preserving maps between lattices
  • A function $f: L_1 \to L_2$ is a lattice homomorphism if for any $a, b \in L_1$, $f(a \vee b) = f(a) \vee f(b)$ and $f(a \wedge b) = f(a) \wedge f(b)$
  • Lattice homomorphisms preserve the join and meet operations
• Lattice isomorphisms: Bijective lattice homomorphisms
  • Two lattices $L_1$ and $L_2$ are isomorphic if there exists a bijective lattice homomorphism $f: L_1 \to L_2$
  • Isomorphic lattices have the same structure and properties
• Lattice embeddings: Injective lattice homomorphisms
  • A lattice embedding is an injective lattice homomorphism $f: L_1 \to L_2$
  • If there exists a lattice embedding from $L_1$ to $L_2$, then $L_1$ is isomorphic to a sublattice of $L_2$
• Quotient lattices: Lattices obtained by collapsing elements of a lattice based on a congruence relation
  • A congruence relation on a lattice $L$ is an equivalence relation that respects the join and meet operations
  • The quotient lattice $L/\sim$ is the set of equivalence classes under the congruence relation, with the induced join and meet operations

Applications of Lattices

• Lattices in computer science:
  • Lattices are used to model data structures and algorithms
  • Examples include the lattice of subsets (power set) and the lattice of partitions
  • Lattices are also used in programming language semantics and type theory
• Lattices in cryptography:
  • Lattices are used in the construction of cryptographic schemes
  • Lattice-based cryptography is believed to be secure against quantum computers
  • Examples include the Learning with Errors (LWE) problem and the Short Integer Solution (SIS) problem
• Lattices in linguistics:
  • Lattices are used to model hierarchical structures in language
  • Examples include the lattice of phonemes and the lattice of syntactic categories
  • Lattices can also be used to represent semantic relationships between words and concepts
• Lattices in physics:
  • Lattices are used to model crystal structures and their properties
  • Examples include the Bravais lattices and the reciprocal lattices
  • Lattices are also used in the study of phase transitions and critical phenomena
• Lattices in social sciences:
  • Lattices are used to model hierarchical structures in social networks and organizations
  • Examples include the lattice of subgroups of a group and the lattice of concepts in formal concept analysis
  • Lattices can also be used to represent preference relations and decision-making processes

Common Theorems and Proofs

• Duality Principle: Every theorem about lattices has a dual theorem obtained by interchanging joins and meets
  • Proof: The duality principle follows from the symmetry of the lattice axioms with respect to joins and meets
• Birkhoff's Representation Theorem: Every finite distributive lattice is isomorphic to the lattice of down-sets of a poset
  • Proof: The proof constructs an isomorphism between the given finite distributive lattice and the lattice of down-sets of its join-irreducible elements
• Fundamental Theorem of Finite Distributive Lattices: Every finite distributive lattice is isomorphic to a sublattice of a power set lattice
  • Proof: This theorem follows from Birkhoff's Representation Theorem and the fact that the lattice of down-sets of a poset is a sublattice of the power set lattice of the poset
• Modular Law: In a modular lattice, if $a \leq c$, then $(a \vee b) \wedge c = a \vee (b \wedge c)$
  • Proof: The proof uses the modular inequality and the properties of joins and meets to establish the equality
• Lattice Isomorphism Theorems:
  • First Isomorphism Theorem: If $f: L_1 \to L_2$ is a surjective lattice homomorphism, then $L_1/\ker(f) \cong L_2$
  • Second Isomorphism Theorem: If $S$ is a sublattice of $L$ and $I$ is an ideal of $L$, then $S/(S \cap I) \cong (S \vee I)/I$
  • Third Isomorphism Theorem: If $I$ and $J$ are ideals of $L$ with $I \subseteq J$, then $(L/I)/(J/I) \cong L/J$
  • Proofs: The proofs of these theorems follow similar techniques to the isomorphism theorems for groups and rings, using the properties of lattice homomorphisms and quotient lattices

Exercises and Problem-Solving

• Determine whether the given sets with the specified operations form a lattice:
  • The set of real numbers with the operations $\max$ and $\min$
  • The set of positive integers with the operations $\gcd$ and $\operatorname{lcm}$
  • The set of subspaces of a vector space with the operations of intersection and sum
• Prove that the lattice of divisors of a positive integer is distributive
• Find the join and meet of the given elements in the specified lattice:
  • Elements 2 and 3 in the lattice of divisors of 12
  • Elements $\{1, 2\}$ and $\{2, 3\}$ in the power set lattice of $\{1, 2, 3\}$
• Draw the Hasse diagram of the lattice of subsets of $\{a, b, c\}$
• Prove that a finite lattice is modular if and only if it does not contain a sublattice isomorphic to the pentagon lattice $N_5$
• Determine whether the given function is a lattice homomorphism:
  • $f: \mathcal{P}(\{1, 2, 3\}) \to \mathcal{P}(\{a, b\})$ defined by $f(A) = \{a\}$ if $1 \in A$ and $f(A) = \{b\}$ otherwise
  • $f: \mathbb{N} \to \mathbb{N}$ defined by $f(n) = n^2$, where $\mathbb{N}$ is the lattice of natural numbers under divisibility
• Find the sublattice generated by the given elements in the specified lattice:
  • Elements 2 and 4 in the lattice of divisors of 24
  • Elements $\{1\}$ and $\{2, 3\}$ in the power set lattice of $\{1, 2, 3\}$
• Construct the quotient lattice of the given lattice by the specified congruence relation:
  • The lattice of divisors of 12 with the congruence relation $a \sim b$ if and only if $a + b$ is even
  • The power set lattice of $\{1, 2, 3\}$ with the congruence relation $A \sim B$ if and only if $|A| = |B|$