distributivity in po-groups
Let be a po-group and be a set of elements of . Denote the supremum of elements of , if it exists, by . Similarly, denote the infimum of elements of , if it exists, by . Furthermore, let , and for any , let and .
If exists, so do and .
If 1. is true, then .
exists iff exists; when this is the case, .
If exists, so do , and .
If 4. is true, then .
If 1. is true and , then exists and is equal to .
(1. and 2.) Clearly, for each , , so that , and therefore elements of are bounded from above by . To show that is the least upper bound of elements of , suppose is the upper bound of elements of , that is, for all , this means that for all . Since is the least upper bound of the ’s, , so that . This shows that is the supremum of elements of ; in other words, . Similarly, exists and as well.
(4. and 5.) This is just the dual of 1. and 2., so the proof is omitted.
This completes the proof. ∎
One can use this result to prove the following: every Dedekind complete po-group is an Archimedean po-group.
Suppose for all integers . Let . Then is bounded from above by so has least upper bound . Then , since . As a result, multiplying both sides by , we get . ∎
|Title||distributivity in po-groups|
|Date of creation||2013-03-22 17:05:12|
|Last modified on||2013-03-22 17:05:12|
|Last modified by||CWoo (3771)|