Difference between revisions of "Archimedean semi-group"
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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 | + | {{TEX|done}} |
| + | 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: | ||
| − | + | $$ab=\begin{cases}a+b&\text{if }a+b\leq1,\\\infty&\text{if }a+b>1.\end{cases}$$ | |
| − | The former case occurs if and only if | + | The former case occurs if and only if $S$ is a semi-group with cancellation. |
====References==== | ====References==== | ||
| Line 10: | Line 11: | ||
''O.A. Ivanova'' | ''O.A. Ivanova'' | ||
| − | A semi-group | + | 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|[1]]]. 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|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. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.S. Putcha, "Band of | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.S. Putcha, "Band of $t$-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> |
Revision as of 11:49, 9 November 2014
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.
References
| [1] | L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) |
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 [1]. 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 [2], [3], was given for the commutative case only.
References
| [1] | M.S. Putcha, "Band of $t$-Archimedean semigroups" Semigroup Forum , 6 (1973) pp. 232–239 |
| [2] | A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) |
| [3] | 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=34406
Archimedean semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Archimedean_semi-group&oldid=34406
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article