User:Dfeuer/sandbox

NBG Set Theory

 * User:Dfeuer/Definition:Class
 * User:Dfeuer/Definition:Set
 * User:Dfeuer/Definition:Subclass
 * User:Dfeuer/Definition:First-Order Formula
 * User:Dfeuer/Axiom of Extensionality
 * User:Dfeuer/Axiom Schema of Separation
 * User:Dfeuer/Not Every Class is a Set
 * User:Dfeuer/Each Class has Subclass which is not Element
 * User:Dfeuer/Definition:Transitive Class
 * User:Dfeuer/Definition:Swelled Class
 * User:Dfeuer/Definition:Supercomplete Class
 * User:Dfeuer/Set is Class
 * User:Dfeuer/Subclass of Set is Set
 * User:Dfeuer/Universal Class is Supercomplete
 * User:Dfeuer/Universal Class is not Set
 * User:Dfeuer/Empty Class Exists and is Unique
 * User:Dfeuer/Empty Class is Subclass of Every Class
 * User:Dfeuer/Empty Class is Supercomplete
 * User:Dfeuer/Axiom of the Empty Set
 * User:Dfeuer/Nonempty Universe implies Axiom of Empty Set
 * User:Dfeuer/Definition:Singleton
 * User:Dfeuer/Set has Unique Singleton
 * User:Dfeuer/Empty Set does not Equal its Singleton
 * User:Dfeuer/Singleton Empty Set is Supercomplete
 * User:Dfeuer/Singletons are Equal iff Elements are Equal
 * User:Dfeuer/Definition:Unordered Pair
 * User:Dfeuer/Axiom of Pairing
 * User:Dfeuer/Weak Pairing implies Axiom of Pairing
 * User:Dfeuer/Singleton Set is Set
 * User:Dfeuer/Definition:Ordered Pair
 * User:Dfeuer/Equality of Ordered Pairs implies Equality of Elements


 * User:Dfeuer/Definition:Inductive Class
 * User:Dfeuer/Definition:Natural Number
 * User:Dfeuer/Natural Numbers Satisfy Peano Axioms


 * User:Dfeuer/Double Induction Principle/Naturals
 * User:Dfeuer/Inductive under Mapping
 * User:Dfeuer/Minimally Inductive under Mapping
 * User:Dfeuer/Double Induction Principle


 * User:Dfeuer/Set Difference of Universal Class and Non-Empty Set is not Transitive

Weird notation: S &amp; F use $\subset$ for "proper subset" and $\supset$ for "implies".

Order and Lattice

 * User:Dfeuer/Definition:Complete Meet Subsemilattice
 * User:Dfeuer/Definition:Complete Join Subsemilattice
 * User:Dfeuer/Intersection of Complete Meet Subsemilattices

Group Theory

 * User:Dfeuer/Coset stuff in progress


 * User:Dfeuer/Definition:Normal Submagma
 * User:Dfeuer/Definition:Normal Subset of Group
 * User:Dfeuer/Definition:Normal Submagma of Group

Topology

 * Order Topology on Convex Subset is Subspace Topology
 * User:Dfeuer/Definition:Stone Space
 * User:Dfeuer/Stone's Representation Theorem for Boolean Algebras
 * User:Dfeuer/Stone Space is Topological Space
 * User:Dfeuer/Compact Subspace of Linearly Ordered Space (draft converse idea)
 * User:Dfeuer/Compact Subspace of Linearly Ordered Space strengthened
 * User:Dfeuer/Compact Subspace of Linearly Ordered Space/Converse Proof 2
 * User:Dfeuer/Closed Set in Linearly Ordered Space
 * User:Dfeuer/Definition:Convex Component (Order Theory)
 * User:Dfeuer/Convex Component is Closed
 * User:Dfeuer/Convex Component of Open Set in GO-Space is Open
 * User:Dfeuer/Compact Subspace of Linearly Ordered Space/Revproof3
 * User:Dfeuer/GO-Space Embeds Densely in LOTS

Usual Topology

 * User:Dfeuer/Definition:Usual Topology

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


 * User:Dfeuer/Transitive Closure of Relation Compatible with Operation is Compatible

Properties of Ordered Groups

 * 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.

So far I believe I've established the equivalence of the theories of transitive relations compatible with group operations and of "cones" compatible with group operations. There is more to flesh out in this realm, as below. I have just started working on the ring stuff.

Discussion point: the notion of a "cone" compatible with an operation can be defined for any magma and closed ringoid. However, without a group structure, I'm not sure if there's a way to link such to compatible relations. Does anyone else see a way to do so?

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
 * User:Dfeuer/Cone Condition Equivalent to Reflexivity

Unstarted/Unfinished Ideas/Conjectures

 * User:Dfeuer/Cone Condition Equivalent to Irreflexivity
 * User:Dfeuer/Cone Condition Equivalent to Symmetry
 * User:Dfeuer/Cone Condition Equivalent to Asymmetry
 * User:Dfeuer/Cone Condition Equivalent to Trichotomy -- I'm not sure this one goes anywhere, since "trichotomy" takes on different flavors for weak and strict orders, and this may confuse matters more than clarify them. For now, the trichotomy bit is rolled into the strict and weak total cone definitions.
 * User:Dfeuer/Definition:Positive Cone
 * User:Dfeuer/Definition:Strict Positive Cone
 * User:Dfeuer/Definition:Total Positive Cone
 * User:Dfeuer/Definition:Strict Total Positive Cone
 * User:Dfeuer/Cone Condition Equivalent to Congruence


 * User:Dfeuer/Definition:Relation Compatible with Closed Ringoid with Zero (or just limit to rigs or rings for the sake of a shorter name?)
 * User:Dfeuer/Definition:Cone Compatible with Closed Ringoid
 * User:Dfeuer/Cone Compatible with Ring Induces Transitive Compatible Relation
 * User:Dfeuer/Transitive Relation Compatible With Ring Induced by Unique Cone
 * User:Dfeuer/Multiplying Compatible Relationship by Zero-Related Element

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