Archimedean semi-group

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