Equational axioms An axiom of an algebraic structure often has the form of an
identity, that is, an
equation such that the two sides of the
equals sign are
expressions that involve operations of the algebraic structure and
variables. If the variables in the identity are replaced by arbitrary elements of the algebraic structure, the equality must remain true. Here are some common examples. ;
Commutativity: An operation * is
commutative if x*y=y*x for every and in the algebraic structure. ;
Associativity: An operation * is
associative if (x*y)*z=x*(y*z) for every , and in the algebraic structure. ;
Left distributivity: An operation * is
left-distributive with respect to another operation + if x*(y+z)=(x*y)+(x*z) for every , and in the algebraic structure (the second operation is denoted here as +, because the second operation is addition in many common examples). ;
Right distributivity: An operation * is
right-distributive with respect to another operation + if (y+z)*x=(y*x)+(z*x) for every , and in the algebraic structure. ;
Distributivity: An operation * is
distributive with respect to another operation + if it is both left-distributive and right-distributive. If the operation * is commutative, left and right distributivity are both equivalent to distributivity.
Existential axioms Some common axioms contain an
existential clause. In general, such a clause can be avoided by introducing further operations, and replacing the existential clause by an identity involving the new operation. More precisely, let us consider an axiom of the form
"for all there is such that where is a -
tuple of variables. Choosing a specific value of for each value of defines a function \varphi:X\mapsto y, which can be viewed as an operation of
arity , and the axiom becomes the identity f(X,\varphi(X))=g(X,\varphi(X)). The introduction of such auxiliary operation complicates slightly the statement of an axiom, but has some advantages. Given a specific algebraic structure, the proof that an existential axiom is satisfied consists generally of the definition of the auxiliary function, completed with straightforward verifications. Also, when computing in an algebraic structure, one generally uses explicitly the auxiliary operations. For example, in the case of
numbers, the
additive inverse is provided by the unary minus operation x\mapsto -x. Also, in
universal algebra, a
variety is a class of algebraic structures that share the same operations, and the same axioms, with the condition that all axioms are identities. What precedes shows that existential axioms of the above form are accepted in the definition of a variety. Here are some of the most common existential axioms. ;
Identity element :A
binary operation * has an identity element if there is an element such that x*e=x\quad \text{and} \quad e*x=x for all in the structure. Here, the auxiliary operation is the operation of arity zero that has as its result. ;
Inverse element :Given a binary operation * that has an identity element , an element is
invertible if it has an inverse element, that is, if there exists an element \operatorname{inv}(x) such that \operatorname{inv}(x)*x=e \quad \text{and} \quad x*\operatorname{inv}(x)=e.For example, a
group is an algebraic structure with a binary operation that is associative, has an identity element, and for which all elements are invertible.
Non-equational axioms The axioms of an algebraic structure can be any
first-order formula, that is a formula involving
logical connectives (such as
"and",
"or" and
"not"), and
logical quantifiers (\forall, \exists) that apply to elements (not to subsets) of the structure. Such a typical axiom is inversion in
fields. This axiom cannot be reduced to axioms of preceding types. (it follows that fields do not form a
variety in the sense of
universal algebra.) It can be stated:
"Every nonzero element of a field is invertible;" or, equivalently:
the structure has a unary operation such that :\forall x, \quad x=0 \quad\text{or} \quad x \cdot\operatorname{inv}(x)=1. The operation can be viewed either as a
partial operation that is not defined for ; or as an ordinary function whose value at 0 is arbitrary and must not be used. == Common algebraic structures ==