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Definition
Totally ordered commutative abelian rings groups
A An commutative abelian ring group is a totally ordered commutative abelian ring group 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 or implies
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for all elements and , or
Totally Strictly ordered commutative pointed abelian groups-algebras
Given A totally ordered commutative rings ring and is a strictly ordered pointed abelian group if it comes with an element , and a commutative type ring family homomorphism is such monotonic that 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.
- 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
A totally ordered commutative ring is a totally ordered commutative -algebra if there is a function such that for all positive integers , .
Strictly ordered pointed -vector space
Strictly ordered integral ring
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A totally ordered commutative ring is a strictly ordered integral ring if it comes with a type family such that
Archimedean ordered pointed -vector space
- 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
A strictly ordered pointed -vector space is an Archimedean ordered pointed -vector space if for all elements and , if , then there merely exists a rational number such that and .
Strictly Sequentially Cauchy complete Archimedean ordered integral pointed -algebras -vector space
Given Let strictly ordered integral rings and be an Archimedean ordered pointed , -vector a space monotonic and commutative let 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 be strictly the ordered positive integral elements ring in . is a strictly ordered integral -algebra if there is a functionsequentially Cauchy complete such if that every for Cauchy all sequence positive in integers , converges:.
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: