User:Dfeuer/sandbox

Order Completion
Note: I'm guessing these are true, and will attempt to prove them, but I do not yet know that either of them is.
 * User:Dfeuer/Order Completion is Minimal
 * User:Dfeuer/Minimal Enclosing Complete Lattice is Order Completion

Properties of Compatible Relations

 * Operating on Transitive Relationships Compatible with Operation
 * Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements


 * Properties of Relation Compatible with Group Operation
 * User:Dfeuer/CTR5

Properties of Ordered Groups

 * User:Dfeuer/Properties of Ordered Group
 * User:Dfeuer/OG1
 * User:Dfeuer/OG2
 * User:Dfeuer/OG3
 * User:Dfeuer/OG4
 * Operating on Ordered Group Inequalities
 * User:Dfeuer/OG5


 * User:Dfeuer/Totally Ordered Group with Order Topology is Topological Group

Systematic development of positivity
The names are made up, but the stories are real.

Discussion point: the notion of a "cone" compatible with an operation can be defined for any magma. I suspect that can be done also for closed ringoids generally. However, without a group structure, building a relation from the cone gets somewhat arbitrary and I don't know if there are any other sorts of interesting results about such cones. Should I then restrict the notion only to groups, or should I leave it wide open in case someone finds something interesting in the more general setting? --Dfeuer (talk) 23:02, 30 January 2013 (UTC)

Proofs sketched out reasonably well; cleanup work appreciated

 * User:Dfeuer/Definition:Cone Compatible with Operation
 * User:Dfeuer/Cone Compatible with Group Induces Transitive Compatible Relation
 * User:Dfeuer/Transitive Relation Compatible with Group Operation Induced by Unique Cone
 * User:Dfeuer/Cone Condition Equivalent to Antisymmetry

Unstarted/Unfinished Ideas/Conjectures

 * User:Dfeuer/Cone Condition Equivalent to Reflexivity
 * User:Dfeuer/Cone Condition Equivalent to Irreflexivity
 * User:Dfeuer/Cone Condition Equivalent to Symmetry
 * User:Dfeuer/Cone Condition Equivalent to Asymmetry
 * User:Dfeuer/Definition:Positive Cone
 * User:Dfeuer/Definition:Strict Positive Cone
 * User:Dfeuer/Definition:Total Positive Cone
 * User:Dfeuer/Definition:Total Strict Positive Cone


 * User:Dfeuer/Definition:Cone Compatible with Closed Ringoid
 * User:Dfeuer/Cone Compatible with Ring Induces Transitive Compatible Relation
 * User:Dfeuer/Transitive Compatible Relation on Ring Induced by Unique Cone


 * User:Dfeuer/OR-musings

Properties of Ordered Rings

 * Product of Positive Element and Element Greater than One
 * User:Dfeuer/Strictly Positive Power of Strictly Positive Element Greater than One Not Less than Element
 * User:Dfeuer/OR1
 * User:Dfeuer/OR2
 * User:Dfeuer/OR3
 * User:Dfeuer/OR4
 * User:Dfeuer/OR5
 * User:Dfeuer/OR6
 * User:Dfeuer/OR7
 * User:Dfeuer/OR8
 * User:Dfeuer/OR9
 * User:Dfeuer/OR10
 * User:Dfeuer/OR11


 * User:Dfeuer/Totally Ordered Division Ring with Order Topology is Topological Division Ring
 * User:Dfeuer/Totally Ordered Field with Order Topology is Topological Field

Lexicographic Orderings

 * User:Dfeuer/Definition:Lexicographic Ordering on Product
 * User:Dfeuer/Definition:Lexicographic Ordering of Finite Sequences

p-norms

 * User:Dfeuer/Derivative of P-Norm wrt P
 * User:Dfeuer/P-Norm of Real Sequence is a Strictly Decreasing Function of P
 * User:Dfeuer/Bounds for P-Norm of Real Sequence

Useful links
Axiom of Foundation at NLab