A function
is an -ary
operation on ; is its ``arity.''
(For we say ``nullary'', ``unary'', ``binary'', ``ternary''.)
An algebra is a set with a given family of operations
(
), called the ``basic operations'' of .
Officially, the algebra is
.
Texts often use a separate letter to distinguish the algebra from the set,
but we'll follow the informal practice of group theory and use
for both.
The type of
is the function
given by
, the arity of
.
Two algebras of the same type are similar. In discussions
involving more than one algebra, we'll normally assume that all
the algebras are similar. Usually will be finite; if
, then it is simplest to choose
and write the
as a sequence.
For example, the type of a Boolean algebra
can be written
.
A subalgebra of an algebra is a subset
that
is closed under all the basic operations of .
An algebra is said to be generated by its
elements
if the smallest subalgebra of that contains all the is itself.
A homomorphism between similar algebras is a map
compatible with the basic operations of and .
The direct product of a family of similar algebras,
or more generally
, is the set-theoretic cartesian product
with operations computed coordinatewise.
A congruence relation on is an equivalence relation
on that is compatible with the basic operations of .
For a congruence relation on , the blocks of
form an algebra of the same type, with a
natural surjective homomorphism
.