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 
.
