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A totally [[Ordered semi-group|ordered semi-group]] all strictly-positive (strictly-negative) elements of which belong to the same [[Archimedean class|Archimedean class]]. All naturally ordered Archimedean semi-groups (cf. [[Naturally ordered groupoid|Naturally ordered groupoid]]) are isomorphic to some sub-semi-group of one of the following semi-groups: the additive semi-group of all non-negative real numbers; the semi-group of all real numbers in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a0131401.png" /> with the usual order and with the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a0131402.png" />; the semi-group consisting of all real numbers in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a0131403.png" /> and the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a0131404.png" /> with the usual order and with the operations:
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A totally [[Ordered semi-group|ordered semi-group]] all strictly-positive (strictly-negative) elements of which belong to the same [[Archimedean class|Archimedean class]]. All naturally ordered Archimedean semi-groups (cf. [[Naturally ordered groupoid|Naturally ordered groupoid]]) are isomorphic to some sub-semi-group of one of the following semi-groups: the additive semi-group of all non-negative real numbers; the semi-group of all real numbers in the interval $(0,1)$ with the usual order and with the operation $ab=\min\{a+b,1\}$; the semi-group consisting of all real numbers in the interval $(0,1)$ and the symbol $\infty$ with the usual order and with the operations:
  
<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/a/a013/a013140/a0131405.png" /></td> </tr></table>
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$$ab=\begin{cases}a+b&\text{if }a+b\leq1,\\\infty&\text{if }a+b>1.\end{cases}$$
  
The former case occurs if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a0131406.png" /> is a semi-group with cancellation.
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The former case occurs if and only if $S$ is a [[semi-group with cancellation]].
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>
 
  
 
''O.A. Ivanova''
 
''O.A. Ivanova''
  
A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a0131407.png" /> which satisfies the following condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a0131408.png" /> there exists a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a0131409.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314012.png" />), the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314013.png" /> is called left (right) Archimedean. For commutative semi-groups all these concepts are equivalent. Any commutative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314014.png" /> is uniquely decomposable into a band of Archimedean semi-groups (and such a decomposition coincides with the finest decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314015.png" /> into a band of semi-groups). This result may be generalized in a different manner to non-commutative semi-groups [[#References|[1]]]. A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314016.png" /> with an idempotent is Archimedean (right Archimedean) if and only if it has a kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314017.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314018.png" /> contains an idempotent (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314019.png" /> is a [[Right group|right group]], cf. also [[Kernel of a semi-group|Kernel of a semi-group]]) while the Rees quotient semi-group (cf. [[Semi-group|Semi-group]]) is a [[Nil semi-group|nil semi-group]]. Archimedean semi-groups without idempotents are harder to study. A complete description in terms of certain constructions, which is especially clear for semi-groups with a cancellation law [[#References|[2]]], [[#References|[3]]], was given for the commutative case only.
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A semi-group $S$ which satisfies the following condition: For any $a,b\in S$ there exists a natural number $n$ such that $a^n\in SbS$. If $a^n\in Sb$ ($a^n\in bS$), the semi-group $S$ is called left (right) Archimedean. For commutative semi-groups all these concepts are equivalent. Any commutative semi-group $S$ is uniquely decomposable into a band of Archimedean semi-groups (and such a decomposition coincides with the finest decomposition of $S$ into a band of semi-groups). This result may be generalized in a different manner to non-commutative semi-groups [[#References|[a1]]]. A semi-group $S$ with an idempotent is Archimedean (right Archimedean) if and only if it has a kernel $K$ and if $K$ contains an idempotent ($K$ is a [[Right group|right group]], cf. also [[Kernel of a semi-group]]) while the Rees quotient semi-group (cf. [[Semi-group]]) is a [[nil semi-group]]. Archimedean semi-groups without idempotents are harder to study. A complete description in terms of certain constructions, which is especially clear for semi-groups with a cancellation law [[#References|[a2]]], [[#References|[a3]]], was given for the commutative case only.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.S. Putcha,  "Band of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013140/a01314020.png" />-Archimedean semigroups"  ''Semigroup Forum'' , '''6'''  (1973)  pp. 232–239</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T. Tamura,  "Construction of trees and commutative Archimedean semigroups"  ''Math. Nachr.'' , '''36''' :  5–6  (1968)  pp. 255–287</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963) {{ZBL|0137.02001}}</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Putcha,  "Band of $t$-Archimedean semigroups"  ''Semigroup Forum'' , '''6'''  (1973)  pp. 232–239</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Tamura,  "Construction of trees and commutative Archimedean semigroups"  ''Math. Nachr.'' , '''36''' :  5–6  (1968)  pp. 255–287</TD></TR>
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</table>
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[[Category:Order, lattices, ordered algebraic structures]]

Latest revision as of 09:17, 2 April 2023

A totally ordered semi-group all strictly-positive (strictly-negative) elements of which belong to the same Archimedean class. All naturally ordered Archimedean semi-groups (cf. Naturally ordered groupoid) are isomorphic to some sub-semi-group of one of the following semi-groups: the additive semi-group of all non-negative real numbers; the semi-group of all real numbers in the interval $(0,1)$ with the usual order and with the operation $ab=\min\{a+b,1\}$; the semi-group consisting of all real numbers in the interval $(0,1)$ and the symbol $\infty$ with the usual order and with the operations:

$$ab=\begin{cases}a+b&\text{if }a+b\leq1,\\\infty&\text{if }a+b>1.\end{cases}$$

The former case occurs if and only if $S$ is a semi-group with cancellation.


O.A. Ivanova

A semi-group $S$ which satisfies the following condition: For any $a,b\in S$ there exists a natural number $n$ such that $a^n\in SbS$. If $a^n\in Sb$ ($a^n\in bS$), the semi-group $S$ is called left (right) Archimedean. For commutative semi-groups all these concepts are equivalent. Any commutative semi-group $S$ is uniquely decomposable into a band of Archimedean semi-groups (and such a decomposition coincides with the finest decomposition of $S$ into a band of semi-groups). This result may be generalized in a different manner to non-commutative semi-groups [a1]. A semi-group $S$ with an idempotent is Archimedean (right Archimedean) if and only if it has a kernel $K$ and if $K$ contains an idempotent ($K$ is a right group, cf. also Kernel of a semi-group) while the Rees quotient semi-group (cf. Semi-group) is a nil semi-group. Archimedean semi-groups without idempotents are harder to study. A complete description in terms of certain constructions, which is especially clear for semi-groups with a cancellation law [a2], [a3], was given for the commutative case only.

References

[1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) Zbl 0137.02001
[a1] M.S. Putcha, "Band of $t$-Archimedean semigroups" Semigroup Forum , 6 (1973) pp. 232–239
[a2] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[a3] T. Tamura, "Construction of trees and commutative Archimedean semigroups" Math. Nachr. , 36 : 5–6 (1968) pp. 255–287
How to Cite This Entry:
Archimedean semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Archimedean_semi-group&oldid=16531
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article