# 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.

• Cor­rect me if I’m wrong, but isn’t it idiosyn­cratic to define $$\leq$$ as a pred­i­cate rather than a re­la­tion? I know of at least three books that de­scribe it as a re­la­tion: The Joy of Sets by Devlin, Prin­ci­ples of Math­e­mat­i­cal Anal­y­sis by Rudin, and In­tro­duc­tion to Lat­tice and Order by Davey and Priestly.

Also, isn’t $$\leq$$ called an or­der rather than a com­par­i­son?

I bring this up be­cause I would like there to be con­sis­tency be­tween this page and the Par­tially or­dered set page. I think both pages should fol­low the con­ven­tions of math­e­mat­ics.

• The word “bi­nary pred­i­cate” I got from Wikipe­dia’s ar­ti­cle on or­dered fields, but it looks like it redi­rects to “bi­nary re­la­tion” any­way, so I’ll change that.

And “com­par­i­son op­er­a­tor” is the ter­minol­ogy in com­puter sci­ence (or at least the one com­monly used in pro­gram­ming lan­guages); I wasn’t aware that the op­er­a­tor was called an “or­der” in math­e­mat­ics in gen­eral.