Editing Universal algebra (section)

== Varieties ==
{{Main|Variety (universal algebra)}}

A collection of algebraic structures defined by identities is called a [[Variety (universal algebra)|variety]] or '''equational class'''.

Restricting one's study to varieties rules out:
* [[Quantification (logic)|quantification]], including  [[universal quantification]] (∀) except before an equation, and [[existential quantification]] (∃)
* [[logical connective]]s other than [[logical conjunction|conjunction]] (∧)
* [[Finitary relation|relations]] other than equality, in particular [[inequality (mathematics)|inequalities]], both {{nowrap|''a'' ≠ ''b''}} and [[Order theory|order relations]]

The study of equational classes can be seen as a special branch of [[model theory]], typically dealing with structures having operations only (i.e. the [[signature (logic)|type]] can have symbols for functions but not for [[Finitary relation|relations]] other than equality), and in which the language used to talk about these structures uses equations only.

Not all [[algebraic structure]]s in a wider sense fall into this scope. For example, [[ordered group]]s involve an ordering relation, so would not fall within this scope.

The class of [[field (mathematics)|field]]s is not an equational class because there is no type (or "signature") in which all field laws can be written as equations (inverses of elements are defined for all ''non-zero'' elements in a field, so inversion cannot be added to the type).

One advantage of this restriction is that the structures studied in universal algebra can be defined in any [[category theory|category]] that has ''finite [[product (category theory)|product]]s''. For example, a [[topological group]] is just a group in the category of [[topological space]]s.

=== Examples ===
Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way, since the usual definitions often involve quantification or inequalities.

==== 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]]).

==== Other examples ====
Most algebraic structures are examples of universal algebras.
* [[Ring (mathematics)|Rings]], [[semigroup]]s, [[quasigroup]]s, [[groupoid]]s, [[Magma (mathematics)|magmas]], [[Loop (algebra)|loops]], and others.
* [[Vector space]]s over a fixed field and [[module (mathematics)|modules]] over a fixed ring are universal algebras. These have a binary addition and a family of unary scalar multiplication operators, one for each element of the field or ring.

Examples of relational algebras include [[semilattice]]s, [[lattice (order)|lattices]], and [[Boolean algebra]]s.