View source for Ordered group

A [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701201.png" /> with an [[Order relation|order relation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701202.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701203.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701204.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701205.png" /> entails <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701206.png" />. If the order is total (respectively, partial), one speaks of a [[Totally ordered group|totally ordered group]] (respectively, a [[Partially ordered group|partially ordered group]]).

An order homomorphism of a (partially) ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701207.png" /> into an ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701208.png" /> is a [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701209.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012010.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012013.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012015.png" />. The kernels of order homomorphisms are the convex normal subgroups (cf. [[Convex subgroup|Convex subgroup]]; [[Normal subgroup|Normal subgroup]]). The set of right cosets of a totally ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012016.png" /> with respect to a convex subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012017.png" /> is totally ordered by putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012018.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012020.png" /> is a convex normal subgroup of a totally ordered group, then this order relation turns the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012021.png" /> into a totally ordered group.

The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012022.png" /> of convex subgroups of a totally ordered group possesses the following properties: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012023.png" /> is totally ordered by inclusion and closed under intersections and unions; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012024.png" /> is infra-invariant, i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012025.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012026.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012027.png" />; c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012028.png" /> is a jump in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012029.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012031.png" />, and there is no convex subgroup between them, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012032.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012033.png" />, the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012034.png" /> is an [[Archimedean group|Archimedean group]] and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012035.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012036.png" /> is the normalizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012038.png" /> (cf. [[Normalizer of a subset|Normalizer of a subset]]); and d) all subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012039.png" /> are strongly isolated, i.e. for any finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012041.png" /> and any subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012042.png" /> the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012043.png" /></td> </tr></table>

entails <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012044.png" />.

An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012045.png" /> of an ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012046.png" /> by an ordered group (cf. [[Extension of a group|Extension of a group]]) is an ordered group if the order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012047.png" /> is stable under all inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012048.png" />. An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012049.png" /> of an ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012050.png" /> by a finite group is an ordered group if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012051.png" /> is torsion-free and if the order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012052.png" /> is stable under all inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012053.png" />.

The order type of a countable ordered group has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012055.png" /> are the order types of the set of integers and of rational numbers, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012056.png" /> is an arbitrary countable ordinal. Every ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012057.png" /> is a [[Topological group|topological group]] relative to the interval topology, in which a base of open sets consists of the open intervals

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012058.png" /></td> </tr></table>

A [[Convex subgroup|convex subgroup]] of an ordered group is open in this topology.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Kokorin,   V.M. Kopytov,   "Fully ordered groups" , Israel Program Sci. Transl.  (1974)  (Translated from Russian)</TD></TR></table>



====Comments====
If the order relation on the partially ordered group defines a [[Lattice|lattice]] (i.e. for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012059.png" /> there exists a greatest lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012060.png" /> and a least upper bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012061.png" />), then one speaks of a [[Lattice-ordered group|lattice-ordered group]] or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012063.png" />-group; cf. also [[Ordered semi-group|Ordered semi-group]]. These turn up naturally in many branches of mathematics. For a survey of the current state-of-the-art in this field see [[#References|[a1]]]–[[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Anderson,   T. Feil,   "Lattice-ordered groups. An introduction" , Reidel  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M.W. Glass (ed.)  W.Ch. Holland (ed.) , ''Lattice-ordered groups. Advances and techniques'' , Kluwer  (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Martinez (ed.) , ''Ordered algebraic structures'' , Kluwer  (1989)</TD></TR></table>