Definition
Commutative -algebras
Every commutative ring has a commutative ring homomorphism from the integers to , and there is an injection from the positive integers to the integers.
A commutative ring is a commutative -algebra if there is a function such that for all positive integers , .
Totally ordered commutative rings
A commutative ring is a totally ordered commutative ring if it comes with a function such that
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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 , .
Every totally ordered commutative ring has a monotonic commutative ring homomorphism from the integers to , and there is an monotonic injection from the positive integers to the integers.
A totally ordered commutative ring is a totally ordered commutative -algebra if there is a function such that for all positive integers , .
Strictly ordered integral ring
A totally ordered commutative ring is a strictly ordered integral ring if it comes with a type family such that
- for all elements and , is a proposition
- for all elements , is false
- for all elements , , and , if , then or
- for all elements and , if is false and is false, then
- for all elements and , if , then is false.
- 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 .
Every strictly ordered ring has a strictly monotonic commutative ring homomorphism from the integers to , and there is an strictly monotonic injection from the positive integers to the integers.
A strictly ordered integral ring is a strictly ordered integral -algebra if there is a function such that for all positive integers , .
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 -algebras
Let be an Archimedean ordered integral -algebra and let
be the positive elements in . is sequentially Cauchy complete if every Cauchy sequence in converges: