Definition:Naturally Ordered Semigroup
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Definition
A naturally ordered semigroup is a (totally) ordered commutative semigroup $\left({S, \circ, \preceq}\right)$ satisfying:
- NO 1: The set $S$ is well-ordered by $\preceq$.
- NO 2: $\forall m, n, p \in S: m \circ p = n \circ p \iff m = n$.
- NO 3: $\forall m, n \in S: m \preceq n \implies \exists p \in S: m \circ p = n$.
- NO 4: $\exists m, n \in S: m \ne n$.
Comment
The proofwiki editor prime.mover has begun this article and plans to complete it some time in the future.
Still to be done: It can be proved that each of the properties that defines the natural ordered semigroup is independent of the others. You can not drop one and still uniquely define a naturally ordered semigroup, neither can you deduce one from the other three.
It can be found in prime.mover's Stubs.
If you want to contribute to this page yourself, then please leave a note on prime.mover's talk page informing him of your intentions.
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The proofwiki editor prime.mover has begun this article and plans to complete it some time in the future.
Still to be done: It can also be proved that if $\left({S, \circ, \preceq}\right)$ satisfies conditions NO 1, NO 2 and NO 3, then $\circ$ is commutative on $S$. Therefore specifying commutativity is not strictly speaking necessary.
It can be found in prime.mover's Stubs.
If you want to contribute to this page yourself, then please leave a note on prime.mover's talk page informing him of your intentions.
If he does not get back to you within two days, feel free to complete the article yourself.
If you plan to replace a large amount of existing material, then please comment out this material by enclosing it in comment delimiters <!-- --> instead of just deleting it.
Also see
- Results about naturally ordered semigroups can be found here.
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 16$
