User:Dfeuer/sandbox

Reflexive Closure and Reflexive Reduction

 * User:Dfeuer/Reflexive Closure of Transitive Relation is Transitive
 * User:Dfeuer/Reflexive Closure of Antisymmetric Relation is Antisymmetric

Dfeuer/CRG4es of Compatible Relations ==


 * Union of Relations Compatible with Operation is Compatible
 * Intersection of Relations Compatible with Operation is Compatible
 * Inverse of Relation Compatible with Operation is Compatible
 * Definition:Relation Conversely Compatible with Operation
 * Definition:Relation Strongly Compatible with Operation
 * User:Dfeuer/Operating on Transitive Relationships Compatible with Operation
 * User:Dfeuer/Operating Repeatedly on Transitive Relationship Compatible with Operation


 * Complement of Relation Compatible with Group is Compatible
 * Relation Compatible with Group is Strongly Compatible
 * Relation Conversely Compatible with Group is Strongly Compatible


 * User:Dfeuer/CRG1
 * User:Dfeuer/CRG2
 * User:Dfeuer/CRG3
 * User:Dfeuer/CRG4
 * User:Dfeuer/CTR5

Properties of Ordered Groups

 * User:Dfeuer/OG1
 * User:Dfeuer/OG2
 * User:Dfeuer/OG3
 * User:Dfeuer/OG4
 * User:Dfeuer/Operating on Ordered Group Relationships
 * User:Dfeuer/OG5

Properties of Relations Compatible with Rings, and/or certain order-like relations on ringoids
The concept here is to come up with an analogue to the notion of a relation compatible with an operation, but for ring-like structures. Clearly we need at least a ringoid with some notion of positive and negative elements. I'll hold off on choosing the exact structure until I've seen what I need to prove some properties of ordered rings.

User:Dfeuer/Definition:Relation Compatible with Ring

Another (more productive?) idea is to bring positivity axioms to ringoids:

Suppose that (R,+,*) is a ringoid. Suppose that $P$ and $N$ are subsets of $R$. Suppose also that the following positivity axioms hold:
 * If $x,y \in P$ then $x+y,x*y\in P$
 * If $x \in P$ and $y \in N$ then $x*y, y*x \in N$
 * If $x, y \in N$ then $x+y \in N$ and $x*y \in P$

One option:

Suppose now that $R,+$ is a group with identity $0$, and that $0 \in P$.

Let $x \prec y$ iff $y - x \in P$.

What sort of relation is $\prec$?

Another option:

Suppose that $+$ is commutative, and let $x \prec y$ if for some $z \in P$, $x+z = y$.

Properties of Ordered Rings

 * 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

Lexicographic Orderings

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


 * User:Dfeuer/Well-Founded Relation Determines Minimal Elements


 * Product of Positive Element and Element Greater than One


 * User:Dfeuer/Strictly Positive Power of Element Greater than One Not Less than Element

Useful links
Axiom of Foundation at NLab