Totally ordered set

chain, linear order

A partially ordered set in which for any two elements $a$ and $b$ either $a \le b$ or $b \le a$. A subset of a totally ordered set is itself a totally ordered set. Every maximal (minimal) element of a totally ordered set is a largest (smallest) element. An important special case of totally ordered sets are the well-ordered sets (cf. Well-ordered set). Among the subsets of a partially ordered set that are totally ordered sets, a particularly important role is played by a composition sequence. A cut of a totally ordered set $P$ is a partition of it into two subsets $A$ and $B$ such that $A \cup B = P$, $A \cap B$ is empty, $A \subseteq B^\nabla$ and $B \subseteq A^\Delta$, where $$ B^\nabla = \{ x \in P : x \le b \ \text{for all}\ b \in B \} \ , $$ $$ A^\Delta = \{ x \in P : x \ge a \ \text{for all}\ a \in A \} \ . $$ The classes $A$ and $B$ are called the lower and upper classes of the cut. One can distinguish the following types of cuts: a jump — there is a largest element in the lower class and a smallest element in the upper class; a Dedekind cut — there is a largest (smallest) element in the lower (upper) class, but no smallest (largest) element in the upper (lower) class; a gap — there is no largest element in the lower class and no smallest element in the upper class. A totally ordered set is said to be a continuous set if all its cuts are Dedekind cuts. A subset $D$ of a totally ordered set $P$ is said to be dense if every interval of $P$ not reducing to a single element contains elements belonging to $D$. The totally ordered set of real numbers can be characterized as a continuous totally ordered set which has neither a largest nor a smallest element, but which contains a countable dense subset. Every countable totally ordered set is isomorphic to some subset of the totally ordered set of all binary fractions in the interval $[0,1]$. A lattice $L$ is isomorphic to a subset of the totally ordered set of integers if and only if every sublattice of it is a retract.

Comments

Totally ordered sets are also called linearly ordered sets.

References

[1]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[2]	P.S. Aleksandrov, "Einführung in die Mengenlehre und in die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[3]	N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French)
[a1]	W. Sierpiński, "Cardinal and ordinal numbers" , PWN (1958)
[a2]	M.M. Zuckerman, "Sets and transfinite numbers" , Macmillan (1974)