Editing Algebraic structure (section)

=== 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.