View source for Convex subgroup of a partially ordered group

A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104001.png" /> of a [[Po-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104002.png" />-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104003.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104004.png" />,

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Many properties of po-groups can be described in terms of convex subgroups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104006.png" /> is convex subgroup of a po-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104007.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104008.png" /> of right cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104009.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040010.png" /> is partially ordered with the induced order: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040011.png" /> if there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040013.png" />.

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040014.png" /> of all convex subgroups of a totally ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040015.png" /> (cf. [[O-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040016.png" />-group]]) is well investigated. It is a complete chain, i.e., it is closed with respect to join and intersection. The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040017.png" /> is infra-invariant, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040024.png" />. For an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040025.png" /> there exist a maximal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040026.png" /> with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040027.png" />, and a minimal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040028.png" /> with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040029.png" />. The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040030.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040031.png" /> (cf. [[Normal subgroup|Normal subgroup]]) and the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040032.png" /> is order isomorphic to some subgroup of the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040033.png" /> of real numbers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040034.png" /> is the normalizer of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040035.png" /> (cf. also [[Normalizer of a subset|Normalizer of a subset]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040037.png" /> for every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040039.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040040.png" /> denotes the [[Commutator subgroup|commutator subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040042.png" />. If, in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040043.png" />, one can find a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040044.png" /> with the properties listed above, then it is possible to turn <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040045.png" /> into an [[O-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040046.png" />-group]] such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040047.png" /> is the system of convex subgroups for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040048.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040049.png" /> is a locally nilpotent o-group, then the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040050.png" /> is a central system of subgroups.

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040051.png" /> of all convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040053.png" />-subgroups (i.e., subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040054.png" /> that are sublattices of the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040055.png" />) is very important for the description of the structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040056.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040057.png" />. It is a complete sublattice of the subgroup lattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040059.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040060.png" /> is isolated, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040061.png" />. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040062.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040063.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040064.png" /> is the kernel of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040065.png" />-homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040066.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040067.png" />; in that case it is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040068.png" />.

Also very important for describing the structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040069.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040070.png" /> are the prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040072.png" />-subgroups, i.e., the convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040073.png" />-subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040074.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040075.png" /> such that the partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040076.png" /> of right cosets is a chain. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040077.png" />-subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040078.png" /> is prime if and only if it is convex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040079.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040080.png" /> for elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040086.png" /> is maximal with respect to the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040087.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040088.png" /> is prime. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040089.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040090.png" /> is the intersection of a suitable set of prime subgroups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040091.png" />, then there exists a natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040092.png" />-homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040093.png" /> onto a transitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040094.png" />-subgroup of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040095.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040096.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040097.png" /> is the group of order automorphisms of the totally ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040098.png" /> of right cosets.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c11040099.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c110400100.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c110400101.png" />, then the set

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is called a polar and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c110400104.png" />.

This article complements and updates the article [[Convex subgroup|Convex subgroup]] (in Volume 2).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Fuchs,   "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bigard,   K. Keimel,   S. Wolfenstein,   "Groupes et anneaux rétiqulés" , Springer  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  "Lattice-ordered groups: advances and techniques"  A.M.W. Glass (ed.)  W.Ch. Holland (ed.) , Kluwer Acad. Publ.  (1989)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.M. Kopytov,   N.Ya. Medvedev,   "The theory of lattice-ordered groups" , Kluwer Acad. Publ.  (1994)  (In Russian)</TD></TR></table>