# Difference between revisions of "Archimedean group"

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− | A partially [[Ordered group|ordered group]] within which the Archimedean axiom is valid: If $a^n<b$ for all integers $n$ ($a$ and $b$ are elements of the Archimedean group), then $a$ is the unit of the group (in additive notation: If $na<b$ for all integers $n$, then $a=0$). Totally ordered Archimedean groups may be described as follows (Hölder's theorem): A totally ordered group is Archimedean if and only if it is isomorphic to some subgroup of the additive group of real numbers with the natural order. Thus, the additive group of all real numbers is in a certain sense the largest totally ordered Archimedean group. All lattice ordered | + | A partially [[Ordered group|ordered group]] within which the Archimedean axiom is valid: If $a^n<b$ for all integers $n$ ($a$ and $b$ are elements of the Archimedean group), then $a$ is the unit of the group (in additive notation: If $na<b$ for all integers $n$, then $a=0$). Totally ordered Archimedean groups may be described as follows (Hölder's theorem): A totally ordered group is Archimedean if and only if it is isomorphic to some subgroup of the additive group of real numbers with the natural order. Thus, the additive group of all real numbers is in a certain sense the largest totally ordered Archimedean group. All Archimedean [[lattice-ordered group]]s are commutative. Totally ordered groups without non-trivial [[convex subgroup]]s are Archimedean. |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc. (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc. (1967)</TD></TR> | ||

+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> A.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)</TD></TR> | ||

+ | <TR><TD valign="top">[3]</TD> <TD valign="top"> L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) {{ZBL|0137.02001}}</TD></TR> | ||

+ | </table> |

## Latest revision as of 16:45, 4 September 2016

A partially ordered group within which the Archimedean axiom is valid: If $a^n<b$ for all integers $n$ ($a$ and $b$ are elements of the Archimedean group), then $a$ is the unit of the group (in additive notation: If $na<b$ for all integers $n$, then $a=0$). Totally ordered Archimedean groups may be described as follows (Hölder's theorem): A totally ordered group is Archimedean if and only if it is isomorphic to some subgroup of the additive group of real numbers with the natural order. Thus, the additive group of all real numbers is in a certain sense the largest totally ordered Archimedean group. All Archimedean lattice-ordered groups are commutative. Totally ordered groups without non-trivial convex subgroups are Archimedean.

#### References

[1] | G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967) |

[2] | A.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian) |

[3] | L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) Zbl 0137.02001 |

**How to Cite This Entry:**

Archimedean group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Archimedean_group&oldid=31539