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 .

REFERENCES

Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981.

 http://www.thoralf.uwaterloo.ca/htdocs/ualg.html.Grätzer, G. Universal Algebra, 2nd ed. New York: Springer-Verlag, 1979.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1171, 2002.