Editing Algebraic structure (section)

== Common axioms ==
===Equational axioms===
An axiom of an algebraic structure often has the form of an [[identity (mathematics)|identity]], that is, an [[equation (mathematics)|equation]] such that the two sides of the [[equals sign]] are [[expression (mathematics)|expressions]] that involve operations of the algebraic structure and [[variable (mathematics)|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 <math>*</math> is ''commutative'' if <math display = block>x*y=y*x </math> for every {{mvar|x}} and {{mvar|y}} in the algebraic structure.
;[[Associativity]]: An operation <math>*</math> is ''associative'' if <math display = block>(x*y)*z=x*(y*z) </math> for every {{mvar|x}}, {{mvar|y}} and {{mvar|z}} in the algebraic structure.
;[[Left distributivity]]: An operation <math>*</math> is ''left distributive'' with respect to another operation <math>+</math> if <math display = block>x*(y+z)=(x*y)+(x*z) </math> for every {{mvar|x}}, {{mvar|y}} and {{mvar|z}} in the algebraic structure (the second operation is denoted here as <math>+</math>, because the second operation is addition in many common examples).
;[[Right distributivity]]: An operation <math>*</math> is ''right distributive'' with respect to another operation <math>+</math> if <math display = block>(y+z)*x=(y*x)+(z*x) </math> for every {{mvar|x}}, {{mvar|y}} and {{mvar|z}} in the algebraic structure.
;[[Distributivity]]: An operation <math>*</math> is ''distributive'' with respect to another operation <math>+</math> if it is both left distributive and right distributive. If the operation <math>*</math> 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 {{mvar|X}} there is {{mvar|y}} such that'' {{nowrap|<math>f(X,y)=g(X,y)</math>",}} where {{mvar|X}} is a {{mvar|k}}-[[tuple]] of variables. Choosing a specific value of {{mvar|y}} for each value of {{mvar|X}} defines a function <math>\varphi:X\mapsto y,</math> which can be viewed as an operation of [[arity]] {{mvar|k}}, and the axiom becomes the identity <math>f(X,\varphi(X))=g(X,\varphi(X)).</math>

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 [[number]]s, the [[additive inverse]] is provided by the unary minus operation <math>x\mapsto -x.</math>

Also, in [[universal algebra]], a [[variety (universal algebra)|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]] <math>*</math> has an identity element if there is an element {{mvar|e}} such that <math display=block>x*e=x\quad \text{and} \quad e*x=x</math> for all {{mvar|x}} in the structure. Here, the auxiliary operation is the operation of arity zero that has {{mvar|e}} as its result.
;[[Inverse element]]
:Given a binary operation <math>*</math> that has an identity element {{mvar|e}}, an element {{mvar|x}} is ''invertible'' if it has an inverse element, that is, if there exists an element <math>\operatorname{inv}(x)</math> such that <math display=block>\operatorname{inv}(x)*x=e \quad \text{and} \quad x*\operatorname{inv}(x)=e.</math>For example, a [[group (mathematics)|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 logic|first-order formula]], that is a formula involving [[logical connective]]s (such as ''"and"'', ''"or"'' and ''"not"''), and [[logical quantifier]]s (<math>\forall, \exists</math>) that apply to elements (not to subsets) of the structure.

Such a typical axiom is inversion in [[field (mathematics)|fields]]. This axiom cannot be reduced to axioms of preceding types. (it follows that fields do not form a [[variety (universal algebra)|variety]] in the sense of [[universal algebra]].) It can be stated: ''"Every nonzero element of a field is [[invertible element|invertible]];"'' or, equivalently: ''the structure has a [[unary operation]]  {{math|inv}} such that
:<math>\forall x, \quad x=0 \quad\text{or} \quad x \cdot\operatorname{inv}(x)=1.</math>
The operation {{math|inv}} can be viewed either as a [[partial operation]] that is not defined for {{math|1=''x'' = 0}}; or as an ordinary function whose value at 0 is arbitrary and must not be used.