View source for 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|retract]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und in die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Theory of sets" , Addison-Wesley  (1968)  (Translated from French)</TD></TR>
</table>



====Comments====
Totally ordered sets are also called linearly ordered sets.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Sierpiński,   "Cardinal and ordinal numbers" , PWN  (1958)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M.M. Zuckerman,   "Sets and transfinite numbers" , Macmillan  (1974)</TD></TR>
</table>

[[Category:Order, lattices, ordered algebraic structures]]