Showing changes from revision #9 to #10:
Added | Removed | Changed
Definition
Totally Abelian ordered abelian groups
An abelian groupabelian group is a totally pointed ordered set abelian group if it comes with a function with a binary operation called subtraction such that
for all elements,
for all elements , and,
for all elements , and , and,
for all elements , and , and , implies that for all elements,
for all elements and , or
Strictly Halving ordered pointed abelian groups
A totally ordered commutative ringhalving group is a an strictly ordered pointed abelian group if it comes with an elementG and with a type function family such called thathalving or dividing by two such that for all , .
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
Totally ordered halving groups
Strictly ordered pointed -vector space
An abelian group is a totally ordered abelian group 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 , or
Strictly ordered pointed halving groups
A totally ordered commutative ring is a strictly ordered pointed abelian group if it comes with an element and 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
…
Archimedean ordered pointed halving groups-vector space
A strictly ordered pointed halving group -vector space is an Archimedean ordered pointed halving group if for all elements is and an Archimedean ordered pointed -vector , space if for all elements , and then there merely exists a dyadic rational number , if such that , then and there merely exists a rational number such that and .
Sequentially Cauchy complete Archimedean ordered pointed halving groups-vector space
Let be an Archimedean ordered pointed halving group and let-vector space and let
be the positive elements in . is sequentially Cauchy complete if every Cauchy sequence in converges: