🧠Universal Algebra Unit 3 – Subalgebras and Homomorphisms

Key Concepts and Definitions

• Universal algebra studies algebraic structures from a general perspective, focusing on the properties and relationships between different types of algebras
• An algebra $(A, F)$ consists of a non-empty set $A$ and a collection of operations $F$ defined on $A$
• A subalgebra $(B, F|_B)$ of an algebra $(A, F)$ is a subset $B \subseteq A$ closed under all operations in $F$
• A homomorphism $\phi: (A, F) \rightarrow (B, G)$ is a function that preserves the algebraic structure between two algebras
• The kernel of a homomorphism $\phi: (A, F) \rightarrow (B, G)$ is the set $\{(x, y) \in A \times A : \phi(x) = \phi(y)\}$
• A congruence on an algebra $(A, F)$ is an equivalence relation on $A$ that is compatible with all operations in $F$
• Isomorphism theorems describe the relationships between homomorphisms, subalgebras, and quotient algebras

Subalgebras: Formation and Properties

• A subalgebra $(B, F|_B)$ is formed by taking a subset $B$ of the underlying set $A$ and restricting the operations in $F$ to $B$
• For a subset $B \subseteq A$ to be a subalgebra, it must be closed under all operations in $F$
  • If $f \in F$ is an $n$-ary operation and $b_1, \ldots, b_n \in B$, then $f(b_1, \ldots, b_n) \in B$
• The intersection of any collection of subalgebras of an algebra $(A, F)$ is also a subalgebra
• The subalgebra generated by a subset $X \subseteq A$ is the smallest subalgebra containing $X$
  • It can be constructed by repeatedly applying the operations in $F$ to elements of $X$ and their results
• Every algebra $(A, F)$ has at least two subalgebras: the trivial subalgebra $(\emptyset, \emptyset)$ and the algebra $(A, F)$ itself
• Examples of subalgebras include subgroups of groups, subfields of fields, and sublattices of lattices

Homomorphisms: Basics and Types

• A homomorphism $\phi: (A, F) \rightarrow (B, G)$ is a function that preserves the algebraic structure between two algebras
  • For each $n$-ary operation $f \in F$, there exists an $n$-ary operation $g \in G$ such that $\phi(f(a_1, \ldots, a_n)) = g(\phi(a_1), \ldots, \phi(a_n))$ for all $a_1, \ldots, a_n \in A$
• Homomorphisms can be injective (one-to-one), surjective (onto), or bijective (both injective and surjective)
• An isomorphism is a bijective homomorphism, denoted by $\cong$
  • If $(A, F) \cong (B, G)$, then the two algebras have the same algebraic structure
• An endomorphism is a homomorphism from an algebra to itself, i.e., $\phi: (A, F) \rightarrow (A, F)$
• An automorphism is a bijective endomorphism
• Examples of homomorphisms include group homomorphisms, ring homomorphisms, and lattice homomorphisms

Kernels and Congruences

• The kernel of a homomorphism $\phi: (A, F) \rightarrow (B, G)$ is the set $\{(x, y) \in A \times A : \phi(x) = \phi(y)\}$
  • It is an equivalence relation on $A$ that is compatible with all operations in $F$
• A congruence on an algebra $(A, F)$ is an equivalence relation on $A$ that is compatible with all operations in $F$
  • If $\theta$ is a congruence and $f \in F$ is an $n$-ary operation, then $(a_1, b_1), \ldots, (a_n, b_n) \in \theta$ implies $(f(a_1, \ldots, a_n), f(b_1, \ldots, b_n)) \in \theta$
• The kernel of a homomorphism is always a congruence on the domain algebra
• Conversely, every congruence on an algebra $(A, F)$ is the kernel of some homomorphism with domain $(A, F)$
• The set of all congruences on an algebra $(A, F)$ forms a lattice under the inclusion order
• The quotient algebra $(A/\theta, F/\theta)$ can be constructed from an algebra $(A, F)$ and a congruence $\theta$ on $(A, F)$

Isomorphism Theorems

• The First Isomorphism Theorem states that if $\phi: (A, F) \rightarrow (B, G)$ is a homomorphism, then $(A/\ker(\phi), F/\ker(\phi)) \cong \operatorname{Im}(\phi)$
  • This theorem relates the quotient algebra by the kernel of a homomorphism to its image
• The Second Isomorphism Theorem states that if $(B, F|_B)$ is a subalgebra of $(A, F)$ and $\theta$ is a congruence on $(A, F)$, then $(B/(\theta \cap (B \times B)), (F|_B)/(\theta \cap (B \times B))) \cong ((B + \theta)/\theta, (F|_{B + \theta})/\theta)$
  • This theorem relates the quotient of a subalgebra by the restriction of a congruence to the subalgebra formed by the congruence classes intersecting the subalgebra in the quotient algebra
• The Third Isomorphism Theorem states that if $\theta$ and $\psi$ are congruences on an algebra $(A, F)$ with $\theta \subseteq \psi$, then $((A/\theta)/(\psi/\theta), (F/\theta)/(\psi/\theta)) \cong (A/\psi, F/\psi)$
  • This theorem relates the quotient of a quotient algebra to the quotient algebra by the larger congruence
• The Correspondence Theorem establishes a one-to-one correspondence between the congruences on an algebra $(A, F)$ containing a congruence $\theta$ and the congruences on the quotient algebra $(A/\theta, F/\theta)$

Applications in Universal Algebra

• Universal algebra provides a unified framework for studying various algebraic structures, such as groups, rings, lattices, and Boolean algebras
• Birkhoff's HSP Theorem characterizes the classes of algebras that are closed under homomorphic images, subalgebras, and direct products
  • These classes, called varieties, can be defined by a set of identities
• The Freese-McKenzie Commutator Theory studies the relationships between congruences on an algebra using commutators
  • Commutators measure the extent to which two congruences fail to commute
• Tame Congruence Theory classifies the local behavior of finite algebras by analyzing the minimal sets and traces of congruences
  • This theory has applications in the study of constraint satisfaction problems and complexity theory
• Universal algebraic methods have been applied to study the structure and properties of various algebraic objects, such as semigroups, monoids, and clones

Common Examples and Problem-Solving Strategies

• When working with subalgebras, consider the closure properties and generate subalgebras using subsets of the underlying set
• To prove that a function is a homomorphism, verify that it preserves the operations between the two algebras
• When dealing with kernels and congruences, use the compatibility property with respect to the operations
• Apply the isomorphism theorems to relate quotient algebras, subalgebras, and homomorphic images
• Utilize the correspondence between congruences on an algebra and congruences on its quotient algebras
• Analyze the identities satisfied by an algebra to determine its membership in a variety
• Examples of common algebras in universal algebra include:
  • Groups, which are algebras with a binary operation, a unary operation (inverse), and a nullary operation (identity)
  • Rings, which are algebras with two binary operations (addition and multiplication) and two nullary operations (additive and multiplicative identities)
  • Lattices, which are algebras with two binary operations (join and meet) satisfying idempotence, commutativity, associativity, and absorption laws
  • Boolean algebras, which are complemented distributive lattices

Advanced Topics and Current Research

• Mal'cev conditions are a powerful tool for characterizing the properties of varieties using identities involving terms with special properties
  • Examples include the Mal'cev term for congruence permutability and the near unanimity term for congruence distributivity
• The study of the lattice of varieties has led to the development of various techniques, such as the use of critical algebras and the finite basis problem
• The algebraic approach to constraint satisfaction problems (CSPs) has revealed deep connections between the complexity of CSPs and the universal algebraic properties of the associated algebras
• The study of natural dualities has provided a framework for understanding the relationships between algebras and their dual structures, such as Priestley duality for distributive lattices and Stone duality for Boolean algebras
• Recent research in universal algebra has focused on topics such as:
  • The structure and properties of clones and hyperclones
  • The study of algebras with incomplete operations, such as partial algebras and relational structures
  • The application of universal algebraic methods to the study of non-classical logics, such as substructural logics and many-valued logics
  • The investigation of the computational complexity of algebraic structures and their associated decision problems