3.1 Subalgebras and Generated Subalgebras

Subalgebras are subsets of an algebra closed under operations. They're key to understanding algebraic structures, from groups to vector spaces. Subalgebras inherit properties from their parent algebra and form a partial order under set inclusion.

Generated subalgebras are the smallest subalgebras containing a given set of elements. They're crucial for studying algebra structure and lead to concepts like finitely generated algebras. Understanding subalgebras is essential for grasping homomorphisms and free algebras.

Subalgebras in Universal Algebra

Definition and Properties

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• Subalgebra forms a subset of an algebra closed under all operations and contains all constant elements
• Closure property ensures applying any operation to subalgebra elements yields a result within the subalgebra
• Subalgebras inherit algebraic structure and properties from parent algebra (associativity, commutativity, distributivity)
• Intersection of any collection of subalgebras creates another subalgebra
• Every algebra has at least two trivial subalgebras
  • The algebra itself
  • Subalgebra containing only constant elements
• Subalgebras form partial order under set inclusion
  • Smallest subalgebra contains set of constants
  • Largest subalgebra encompasses the entire algebra

Examples and Applications

• In group theory, subgroups serve as subalgebras (cyclic subgroups, normal subgroups)
• For vector spaces, subspaces function as subalgebras (span of vectors, null space)
• In ring theory, subrings act as subalgebras (ideal of a ring, polynomial ring)
• Boolean algebras have Boolean subalgebras (power set algebra, finite Boolean algebras)
• Lattice theory utilizes sublattices as subalgebras (interval sublattice, principal ideal)

Constructing Subalgebras

Process of Construction

• Identify subset of elements from the algebra
• Close subset under all operations of the algebra
• Closure under operation means applying operation to any combination of elements in set yields result within set
• Construct subalgebra by repeatedly applying all operations until no new elements generated
• Process guaranteed to terminate for finite algebras
• May be infinite for algebras with infinitely many elements
• Include all constant elements of algebra in construction process

Techniques and Considerations

• Start with generating set and apply operations systematically
• Use closure properties to identify new elements
• Consider algebraic structure to simplify construction process
• For groups, generate elements by repeated application of group operation
• In vector spaces, use linear combinations of basis vectors
• For rings, include additive and multiplicative closures
• In lattices, consider meets and joins of elements

Proving Subalgebras

Proof Methodology

• Demonstrate closure under all operations
• Verify presence of all constant elements
• Show each operation applied to subset elements yields result within subset
• For unary operations, prove applying operation to any element produces element in subset
• For binary operations, demonstrate operation applied to any pair of elements results in element within subset
• Verify all constant elements of algebra included in subset

Counterexamples and Disproof

• Use counterexamples to disprove subset as subalgebra
• Find operation producing element outside subset when applied to elements within it
• Example in groups: subset ${1, -1}$ of integers under multiplication not a subgroup, as $1 + (-1) = 0$ not in subset
• In vector spaces, set of vectors with only positive components not a subspace
• For rings, set of even integers not a subring of integers, as product of two even integers can be odd

Generated Subalgebras

Concept and Significance

• Generated subalgebra represents smallest subalgebra containing given set of elements
• Denoted as $⟨S⟩$ for generating set S
• Process involves applying all operations to generating set and results until closure achieved
• Play crucial role in studying algebra structure
• Represent minimal algebraic closure of set of elements
• Fundamental in defining free algebras and understanding homomorphisms

Applications and Extensions

• Study relationships between different substructures within algebra
• Leads to concepts like finitely generated algebras
• Addresses generation problem in universal algebra
• In group theory, cyclic groups generated by single element
• Vector spaces use concept of span to generate subspaces
• Polynomial rings generated by variables and coefficients
• In lattice theory, principal ideals generated by single element