Editing Algebraic structure (section)

===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.