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 signature (or type) is a set of function symbols, each assigned a nonnegative integer
arity. 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 
,
 |
A surjective homomorphism 
makes 
a homomorphic image of 
and is an epimorphism. An
epimorphism need not be surjective.
If the homomorphism 
is a bijection, then it is
called an isomorphism. On the class of all algebras,
define a relation 
by 
iff 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 
.
