If A and B are sets, the Cartesian product AB of A and B is defined as the set of all ordered pairs (a, b), with aA and bB. In symbols,
If f: A --> B is a mapping of two sets we write fn for the induced mapping
Definition 6.1. (a) An operation of
rank n (or n-ary operation)
on A is a function
Remark 6.2. (a) Operations of rank 0 on a nonempty set
are functions that have only one value, called constants,
which are identified with their unique values.
Denote by A(n) the set of operations of rank n on A.
Definition 6.3. A type of algebra is a set with a map a: --> N, where N is the set of natural numbers; the elements of are called operators, and if , then a() is called the arity of . If a() = n we also say that is n-ary, and we write
Definition 6.4. Let A be a set and an operator domain; then an -algebra structure F on A is a family of mappings
Definition 6.5. A set A together with an -algebra structure F is called an -algebra.
Given an -algebra A and (n), then applied to an n-tuple (a1, ..., an) from A given an element of A which we write as ( a1a2...an).
Example 6.5.1. If A has only one element, there is only one way of defining an -algebraic structure, because for any integer n, there is only one mapping from An to A. An -algebra with only one element is called trivial; all trivial algebras are isomorphic.
In the following we fix a type of algebra and consider -algebras.
Definition 6.6. Given -algebras A and B, a mapping f: A --> B, and (n), we say the f is compatible with if
Definition 6.7. Consider an -algebra
Remark 6.8. (a) Any subalgebra B of an -algebra
is itself an -algebra
with the basic operations
induced from the basic operations of
such that the inclusion mapping B --> A is a homomorphism
Lemma 6.9. Let A be an -algebra and X a generating set of A. Then any homomorphism of A into another -algebra is completely determined by its restriction to X.
Proof. If t and s are two homomorphisms from A to B which agree on X, the subset A' of all the elements of A at which t and s agree is a subalgebra which contains X. Since X generates A, we have A = A'.
Let X be a set whose elements are called variables.
Let be a type of algebras.
The set T(X) of terms of type
over X is the smallest set such that
For p T(X) we often write p as p(x1,,...,xn) to indicate that the variables occuring in p are among x1,...,xn. A term p is n-ary if the number of variables appearing explicity in p is n.
Given a term p(x1,...,xn) of type
over some set X and given an algebra A of type
we define a mapping pA: An --> A
(2) if p is of the form (p1(x1,...,xn),...,pk(x1,...,xn)), where k, then
pA is the term function on A corresponding to the term p (Often we will drop the superscript A).
Given and X, if T(X) then T(X) is an algebra of type whose fundamenstal operations satisffies
T(X) is called the term agebra of type over X.
Proposition 6.10. (a) T(X) is a free object of
() over X.
Proof. (a) Consider any mapping f: X -->
A from X to an -algebra
A. Then f extends to a mapping f': T(X)
--> A sending each term p(x1,...,xn)
which is an -algebra homomorphism,
and f' is unique as X generates T(X).