Editing Universal algebra (section)

==== Groups ====
As an example, consider the definition of a [[group (mathematics)|group]].  Usually a group is defined in terms of a single binary operation ∗, subject to the axioms:
* [[associative|Associativity]] (as in the [[#Equations|previous section]]):  ''x''&nbsp;∗&nbsp;(''y''&nbsp;∗&nbsp;''z'')&nbsp;&nbsp;=&nbsp; (''x''&nbsp;∗&nbsp;''y'')&nbsp;∗&nbsp;''z''; &nbsp; formally: ∀''x'',''y'',''z''. ''x''∗(''y''∗''z'')=(''x''∗''y'')∗''z''.
* [[Identity element]]:  There exists an element ''e'' such that for each element ''x'', one has ''e''&nbsp;∗&nbsp;''x''&nbsp;&nbsp;=&nbsp; ''x''&nbsp;&nbsp;=&nbsp; ''x''&nbsp;∗&nbsp;''e''; &nbsp; formally: ∃''e'' ∀''x''. ''e''∗''x''=''x''=''x''∗''e''.
* [[Inverse element]]:  The identity element is easily seen to be unique, and is usually denoted by ''e''. Then for each ''x'', there exists an element ''i'' such that ''x''&nbsp;∗&nbsp;''i''&nbsp;&nbsp;=&nbsp; ''e''&nbsp;&nbsp;=&nbsp; ''i''&nbsp;∗&nbsp;''x''; &nbsp; formally: ∀''x''&nbsp;∃''i''.&nbsp;''x''∗''i''=''e''=''i''∗''x''.

(Some authors also use the "[[Closure (mathematics)|closure]]" axiom that ''x''&nbsp;∗&nbsp;''y'' belongs to ''A'' whenever ''x'' and ''y'' do, but here this is already implied by calling ∗ a binary operation.)

This definition of a group does not immediately fit the point of view of universal algebra, because the axioms of the identity element and inversion are not stated purely in terms of equational laws which hold universally "for all&nbsp;..." elements, but also involve the existential quantifier "there exists&nbsp;...".  The group axioms can be phrased as universally quantified equations by specifying, in addition to the binary operation ∗, a nullary operation ''e'' and a unary operation ~, with ~''x'' usually written as ''x''<sup>−1</sup>. The axioms become:
* Associativity:  {{nowrap|1=''x'' ∗ (''y'' ∗ ''z'') &nbsp;=&nbsp;}} {{nowrap|(''x'' ∗ ''y'') ∗ ''z''}}.
* Identity element:  {{nowrap|1=''e'' ∗ ''x'' &nbsp;=&nbsp;}} {{nowrap|1=''x'' &nbsp;=&nbsp;}} {{nowrap|''x'' ∗ ''e''}}; &nbsp; formally: {{nowrap|1=∀''x''. ''e''∗''x''=''x''=''x''∗''e''}}.
* Inverse element:  {{nowrap|1=''x'' ∗ (~''x'') &nbsp;=&nbsp;}} {{nowrap|1=''e'' &nbsp;=&nbsp;}} {{nowrap|(~''x'') ∗ ''x''}}; &nbsp; formally: {{nowrap|1=∀''x''. ''x''∗~''x''=''e''=~''x''∗''x''}}.

To summarize, the usual definition has:
* a single binary operation ([[signature (logic)|signature]] (2))
* 1 equational law (associativity)
* 2 quantified laws (identity and inverse)
while the universal algebra definition has:
* 3 operations: one binary, one unary, and one nullary ([[signature (logic)|signature]] {{nowrap|(2, 1, 0)}})
* 3 equational laws (associativity, identity, and inverse)
* no quantified laws (except outermost universal quantifiers, which are allowed in varieties)

A key point is that the extra operations do not add information, but follow uniquely from the usual definition of a group. Although the usual definition did not uniquely specify the identity element ''e'', an easy exercise shows that it is unique, as is the [[inverse element|inverse]] of each element.

The universal algebra point of view is well adapted to category theory. For example, when defining a [[group object]] in category theory, where the object in question may not be a set, one must use equational laws (which make sense in general categories), rather than quantified laws (which refer to individual elements). Further, the inverse and identity are specified as morphisms in the category. For example, in a [[topological group]], the inverse must not only exist element-wise, but must give a continuous mapping (a morphism). Some authors also require the identity map to be a [[closed inclusion]] (a [[cofibration]]).