List of order structures in mathematics

A set can be ordered in two different senses:

• If its elements do not commute,
• If there is an ordering relation between its elements, in which case the set is either partially or fully ordered.

If a set's elements do not commute, then the elements are enclosed within parentheses instead of brackets. E.g. {a,b} is an unordered pair, so that {a,b}={b,a}, but (a,b) is an ordered pair: $(a,b)\neq (b,a){\mbox{ unless }}a=b.$

E.g. (c,b,a) is an ordered triple different from (a,b,c); (a,d,b,c) is an ordered quadruple different from (a,b,c,d), and (a₁,a₂,a₃, ... , a_n) an ordered n-tuple.

Definition (formal). An ordered pair (a,b) is equivalent to the set {{a},{a,b}}. It can be proven that if {{a},{a,b}} = {{c},{c,d}} then a=c and b=d.