Universal Algebra

Universal algebra studies common properties of all algebraic structures, including groups, rings, fields, lattices, etc.

A universal algebra is a pair , where and are sets and for each , is an operation on . The algebra is finitary if each of its operations is finitary.

A set of function symbols (or operations) of degree is called a signature (or type). Let be a signature. An algebra is defined by a domain (which is called its carrier or universe) and a mapping that relates a function to each -place function symbol from .

Let and be two algebras over the same signature , and their carriers are and , respectively. A mapping is called a homomorphism from to if for every and all ,

If a homomorphism is surjective, then it is called epimorphism. If is an epimorphism, then is called a homomorphic image of . If the homomorphism is a bijection, then it is called an isomorphism. On the class of all algebras, define a relation by if and only if there is an isomorphism from onto . Then the relation is an equivalence relation. Its equivalence classes are called isomorphism classes, and are typically proper classes.

A homomorphism from to is often denoted as . A homomorphism is called an endomorphism. An isomorphism is called an automorphism. The notions of homomorphism, isomorphism, endomorphism, etc., are generalizations of the respective notions in groups, rings, and other algebraic theories.

Identities (or equalities) in algebra over signature have the form

where and are terms built up from variables using function symbols from .

An identity is said to hold in an algebra if it is true for all possible values of variables in the identity, i.e., for all possible ways of replacing the variables by elements of the carrier. The algebra is then said to satisfy the identity .

Wolfram Web Resources