Totally ordered set

A to­tally or­dered set is a pair \((S, \le)\) of a set \(S\) and a to­tal or­der \(\le\) on \(S\), which is a bi­nary re­la­tion that satis­fies the fol­low­ing prop­er­ties:

  1. For all \(a, b \in S\), if \(a \le b\) and \(b \le a\), then \(a = b\). (the an­ti­sym­met­ric prop­erty)

  2. For all \(a, b, c \in S\), if \(a \le b\) and \(b \le c\), then \(a \le c\). (the tran­si­tive prop­erty)

  3. For all \(a, b \in S\), ei­ther \(a \le b\) or \(b \le a\), or both. (the to­tal­ity prop­erty)

A to­tally or­dered set is a spe­cial type of par­tially or­dered set that satis­fies the to­tal prop­erty — in gen­eral, posets only satisfy the re­flex­ive prop­erty, which is that \(a \le a\) for all \(a \in S\).

Ex­am­ples of to­tally or­dered sets

The real num­bers are a to­tally or­dered set. So are any of the sub­sets of the real num­bers, such as the ra­tio­nal num­bers or the in­te­gers.

Ex­am­ples of not to­tally or­dered sets

The com­plex num­bers do not have a canon­i­cal to­tal or­der­ing, and es­pe­cially not a to­tal or­der­ing that pre­serves all the prop­er­ties of the or­der­ing of the real num­bers, al­though one can define a to­tal or­der­ing on them quite eas­ily.

Children:

  • Well-ordered set

    An or­dered set with an or­der that always has a “next el­e­ment”.

Parents:

  • Order theory

    The study of bi­nary re­la­tions that are re­flex­ive, tran­si­tive, and an­ti­sym­metic.