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Idea
I am going to define this in terms of Archimedean ordered Q-algebras…
Definition
Strict order axioms:
Commutative -algebras
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For all terms , is false.
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For all terms , , , implies or
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For all terms , , not and not implies .
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For all terms , , implies not
A commutative ring is a commutative -algebra if there is a commutative ring homomorphism .
Archimedean property:
Totally ordered commutative rings
- For all terms , , and , and implies that there exists a natural number such that , where is the additive -th power (n-fold addition)
A commutative ring is a totally ordered commutative ring if it comes with a function such that
One axioms:
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for all elements ,
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for all elements and ,
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for all elements , , and ,
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for all elements and , implies that for all elements ,
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for all elements and , and implies
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for all elements and , or
Totally ordered commutative -algebras
Given totally ordered commutative rings and , a commutative ring homomorphism is monotonic if for all and , .
A totally ordered commutative ring is a totally ordered commutative -algebra if there is a monotonic commutative ring homomorphism .
Strictly ordered integral ring
A totally ordered commutative ring is a strictly ordered integral ring if it comes with a strict order such that
- for all elements and , if and , then
- for all elements and , if and , then
- for all elements and , if and , then
Strictly ordered integral -algebras
Given strictly ordered integral rings and , a monotonic commutative ring homomorphism is strictly monotonic if for all and , implies .
A strictly ordered integral ring is a strictly ordered integral -algebra if there is a strictly monotonic commutative ring homomorphism .
Archimedean ordered integral -algebras
A strictly ordered integral -algebra is an Archimedean ordered integral -algebra if for all elements and , if , then there merely exists a rational number such that and .
Sequentially Cauchy complete Archimedean ordered integral -algebra
Let be an Archimedean ordered integral -algebra and let
be the positive elements in . is sequentially Cauchy complete if every Cauchy sequence in converges:
See also