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Idea
I am going to define this in terms of Archimedean ordered Q-algebras…
Definition
Commutative -algebras
A commutative ring is a commutative -algebra if there is a commutative ring homomorphism .
Totally ordered commutative rings
A commutative ring is a totally ordered commutative ring if it comes with a function such that
for all elements ,
for all elements and ,
for all elements , , and ,
for all elements and , implies that for all elements ,
for all elements and , and implies
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 type order family such that
for all elements and , is a proposition
for all elements , and , if is false and , then
for all elements , and , if and , 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 .
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: