Editing Universal algebra (section)

== Basic idea ==
{{Main|Algebraic structure}}
{{distinguish|Algebra over a field}}
In universal algebra, an '''{{vanchor|algebra}}''' (or [[algebraic structure]]) is a [[set (mathematics)|set]] ''A'' together with a collection of operations on ''A''.  

=== Arity ===
{{Main|Arity}}
An '''''n''-[[arity|ary]] [[operation (mathematics)|operation]]''' on ''A'' is a [[function (mathematics)|function]] that takes ''n'' elements of ''A'' and returns a single element of ''A''.  Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a ''[[Constant (mathematics)|constant]]'', often denoted by a letter like ''a''.  A 1-ary operation (or ''[[unary operation]]'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~''x''.  A 2-ary operation (or ''[[binary operation]]'') is often denoted by a symbol placed between its arguments (also called [[infix notation]]), like ''x''&nbsp;∗&nbsp;''y''.  Operations of higher or unspecified ''[[arity]]'' are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like ''f''(''x'',''y'',''z'') or ''f''(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>).   One way of talking about an algebra, then, is by referring to it as an [[Outline of algebraic structures#Types of algebraic structures|algebra of a certain type]] <math>\Omega</math>, where <math>\Omega</math> is an ordered sequence of natural numbers representing the arity of the operations of the algebra.  However, some researchers also allow [[infinitary]] operations, such as <math>\textstyle\bigwedge_{\alpha\in J} x_\alpha</math> where ''J'' is an infinite [[index set]], which is an operation in the algebraic theory of [[complete lattice]]s. 

=== Equations ===
After the operations have been specified, the nature of the algebra is further defined by [[axiom]]s, which in universal algebra often take the form of [[Identity (mathematics)#Logic and universal algebra|identities]], or '''equational laws.'''    An example is the [[associative]] axiom for a binary operation, which is given by the equation ''x''&nbsp;∗&nbsp;(''y''&nbsp;∗&nbsp;''z'')&nbsp;= (''x''&nbsp;∗&nbsp;''y'')&nbsp;∗&nbsp;''z''.  The axiom is intended to hold for all elements ''x'', ''y'', and ''z'' of the set ''A''.