Editing Algebraic structure (section)

== Introduction ==

[[Addition]] and [[multiplication]] are prototypical examples of [[operation (mathematics)|operations]] that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, {{math|1=''a'' + (''b'' + ''c'') = (''a'' + ''b'') + ''c''}} and {{math|1=''a''(''bc'') = (''ab'')''c''}} are [[associative law]]s, and {{math|1=''a'' + ''b'' = ''b'' + ''a''}} and {{math|1=''ab'' = ''ba''}} are [[commutative law]]s. Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, the possible moves of an object in three-dimensional space can be combined by performing a first move of the object, and then a second move from its new position. Such moves, formally called [[rigid motion]]s, obey the associative law, but fail to satisfy the commutative law.

Sets with one or more operations that obey specific laws are called ''algebraic structures''. When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem.

In full generality, algebraic structures may involve an arbitrary collection of operations, including operations that combine more than two elements (higher [[arity]] operations) and operations that take only one [[Argument of a function|argument]] ([[unary operation]]s) or even zero arguments ([[nullary operation]]s). The examples listed below are by no means a complete list, but include the most common structures taught in undergraduate courses.