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]]. |
− | |||
− | |||
''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|[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"> | + | <table> |
+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) {{ZBL|0137.02001}}</TD></TR> | ||
+ | <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> | ||
+ | <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> | ||
+ | <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> | ||
+ | </table> | ||
+ | |||
+ | |||
+ | [[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 |
Archimedean semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Archimedean_semi-group&oldid=16531