2.3 Homomorphisms and isomorphisms between structures

Homomorphisms and isomorphisms are crucial tools in model theory, connecting different structures while preserving their essential properties. They allow us to map elements and operations between structures, revealing deep relationships and similarities that might not be immediately obvious.

These structure-preserving maps are fundamental for analyzing and comparing mathematical structures. Homomorphisms maintain basic operations, while isomorphisms establish a perfect correspondence, effectively showing that two structures are identical in their algebraic properties, just with different labels for their elements.

Homomorphisms and Isomorphisms

Defining Structure-Preserving Maps

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• Homomorphisms preserve operations between two algebraic structures of the same type
  • Function f: A → B satisfies f(x * y) = f(x) ∘ f(y) for all x, y in A, where * and ∘ are operations in A and B
  • Maintain algebraic structure but may not preserve all properties
  • Apply to various structures (groups, rings, fields, vector spaces)
• Isomorphisms establish one-to-one correspondence between elements of two structures
  • Bijective homomorphisms with homomorphic inverses
  • Preserve all structural properties
  • Effectively relabel elements between two structures
  • Existence implies structural identity and shared algebraic properties

Characteristics and Examples

• Group homomorphism example: f: (ℤ, +) → (ℤ/2ℤ, +) defined by f(n) = n mod 2
  • Preserves addition: f(a + b) = f(a) + f(b) for all a, b in ℤ
• Ring homomorphism example: f: ℤ → ℤ[i] defined by f(a) = a + 0i
  • Preserves both addition and multiplication
• Isomorphism example: f: ℂ → ℝ² defined by f(a + bi) = (a, b)
  • Bijective and preserves addition and multiplication
• Non-example: f: ℤ → ℤ defined by f(n) = 2n
  • Homomorphism but not isomorphism (not surjective)

Proving Homomorphisms and Isomorphisms

Proving Homomorphisms

• Demonstrate preservation of all defined operations for all elements in the domain
• Utilize structure-specific homomorphism definitions (group, ring)
• For group homomorphisms, show respect for identity element and inverses
  • Example: Prove f: ℤ → ℤ/nℤ defined by f(a) = a mod n is a group homomorphism
    • Show f(a + b) = f(a) + f(b) for all a, b in ℤ
    • Verify f(0) = 0 (identity preservation)
• For ring homomorphisms, prove preservation of addition and multiplication
  • Example: Prove f: ℤ → ℤ[x] defined by f(n) = n is a ring homomorphism
    • Show f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b) for all a, b in ℤ

Proving Isomorphisms

• Prove function is both injective (one-to-one) and surjective (onto)
• Construct inverse function and show both original and inverse are homomorphisms
• Example: Prove f: ℝ → ℝ⁺ defined by f(x) = e^x is an isomorphism
  • Show injectivity: f(x₁) = f(x₂) implies x₁ = x₂
  • Show surjectivity: For any y in ℝ⁺, there exists x in ℝ such that f(x) = y
  • Construct inverse g(y) = ln(y) and prove it's a homomorphism
• Use counterexamples to disprove homomorphism or isomorphism claims
  • Example: f: ℤ → ℤ defined by f(n) = n² is not a group homomorphism
    • Counterexample: f(1 + 2) ≠ f(1) + f(2)

Properties Preserved by Mappings

Homomorphism Preservation

• Algebraic operations (addition, multiplication) defined in structures
• Identity element mapping: f(e_A) = e_B, where e_A and e_B are identities in A and B
• Substructure mapping (subgroups to subgroups, ideals to ideals)
• Element order: order(f(a)) divides order(a) for any element a
• Examples:
  • In group homomorphism f: ℤ → ℤ/nℤ, subgroups of ℤ map to subgroups of ℤ/nℤ
  • For f: (ℤ, +) → (ℤ/2ℤ, +), f(2) has order 1, which divides the order of 2 in ℤ (infinite)

Isomorphism Preservation

• All structural properties including:
  • Order of elements
  • Cyclic nature of groups
  • Abelian property
  • Simplicity of groups
• Topological properties in topological structures
• Vector space dimension
• Examples:
  • Isomorphism between (ℝ, +) and (ℝ⁺, ·) preserves the abelian property
  • Isomorphism f: ℂ → ℝ² preserves 2-dimensional vector space structure

Applications of Homomorphisms and Isomorphisms

Structural Analysis

• Use isomorphisms to prove structural equivalence, applying properties between structures
  • Example: ℤ/nℤ ≅ ℤ/mℤ if and only if n = m, allowing property transfer
• Apply First Isomorphism Theorem to relate groups or rings to homomorphic images
  • G/ker(f) ≅ im(f) for group homomorphism f: G → H
• Study relationships between algebraic structures via homomorphisms
  • Analyze quotient groups and factor rings
• Employ isomorphism theorems for quotient and factor structure analysis
  • Second Isomorphism Theorem: (G/N)/(H/N) ≅ G/H for N ⊴ H ⊴ G

Advanced Applications

• Define and study concepts like normal subgroups and ideals using homomorphisms
  • Normal subgroups as kernels of group homomorphisms
• Classify finite groups up to isomorphism
  • All cyclic groups of order n are isomorphic to ℤ/nℤ
• Apply in representation theory to study abstract structures via linear transformations
  • Use group homomorphisms from abstract groups to groups of matrices
• Utilize in cryptography for designing secure systems
  • RSA encryption uses homomorphic properties of modular exponentiation