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. Maybe. All of this should for now be considered pre-rough-draft material and viewed through a skeptical lens.


 * User:Dfeuer/Definition:Cone Compatible with Semigroup Operation
 * User:Dfeuer/Cone Compatible with Abelian Group Induces Transitive Compatible Relation
 * User:Dfeuer/Transitive Relation Compatible with Abelian Group Operation Induces Unique Cone
 * User:Dfeuer/Cone Condition Equivalent to Antisymmetry
 * User:Dfeuer/Cone Condition Equivalent to Reflexivity


 * User:Dfeuer/Definition:Cone Compatible with Additive Semiring
 * User:Dfeuer/Cone Compatible with Ring Induces Transitive Compatible Relation
 * User:Dfeuer/Transitive Compatible Relation on Ring Induces 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