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 f^{n} for the induced mapping Definition 6.1. (a) An operation of
rank n (or nary operation)
on A is a function
A^{n
}into
A.
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 nary, 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 ntuple (a_{1}, ..., a_{n}) from A given an element of A which we write as ( a_{1}a_{2}...a_{n}). 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 A^{n} 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
A.
Remark 6.8. (a) Any subalgebra B of an algebra
A
is itself an algebra
with the basic operations
induced from the basic operations of
such that the inclusion mapping B > A is a homomorphism
of algebras..
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(x_{1},,...,x_{n}) to indicate that the variables occuring in p are among x_{1},...,x_{n}. A term p is nary if the number of variables appearing explicity in p is n. Given a term p(x_{1},...,x_{n}) of type
over some set X and given an algebra A of type
we define a mapping p^{A}: A^{n} > A
as follows:
(2) if p is of the form (p_{1}(x_{1},...,x_{n}),...,p_{k}(x_{1},...,x_{n})), where _{k}, then p^{A} 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(x_{1},...,x_{n})
to p^{A}(f(x_{1}),...,f(x_{n})),
which is an algebra homomorphism,
and f' is unique as X generates T(X).
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