# Category:Compatible Relations

This category contains results about Compatible Relations.

Definitions specific to this category can be found in Definitions/Compatible Relations.

Let $\struct {S, \circ}$ be a closed algebraic structure.

Let $\mathcal R$ be a relation on $S$.

Then $\mathcal R$ is **compatible with $\circ$** if and only if:

- $\forall x, y, z \in S: x \mathrel {\mathcal R} y \implies \paren {x \circ z} \mathrel {\mathcal R} \paren {y \circ z}$

- $\forall x, y, z \in S: x \mathrel {\mathcal R} y \implies \paren {z \circ x} \mathrel {\mathcal R} \paren {z \circ y}$

## Subcategories

This category has the following 4 subcategories, out of 4 total.

### C

### O

### S

## Pages in category "Compatible Relations"

The following 38 pages are in this category, out of 38 total.

### C

### D

- User:Dfeuer/Cone Compatible with Group Induces Transitive Compatible Relation
- User:Dfeuer/Cone Compatible with Ring Induces Transitive Compatible Relation
- User:Dfeuer/Cone Condition Equivalent to Antisymmetry
- User:Dfeuer/Cone Condition Equivalent to Asymmetry
- User:Dfeuer/Cone Condition Equivalent to Irreflexivity
- User:Dfeuer/Cone Condition Equivalent to Reflexivity
- User:Dfeuer/CTR5
- User:Dfeuer/Definition:Cone Compatible with Operation
- User:Dfeuer/Multiplying Compatible Relationship by Zero-Related Element
- User:Dfeuer/Transitive Relation Compatible with Group Operation Induced by Unique Cone
- Diagonal Complement Relation Compatible with Group Operation
- Diagonal Relation is Universally Compatible

### I

### P

- Preordering of Products under Operation Compatible with Preordering
- Properties of Relation Compatible with Group Operation
- Properties of Relation Compatible with Group Operation/CRG1
- Properties of Relation Compatible with Group Operation/CRG2
- Properties of Relation Compatible with Group Operation/CRG3
- Properties of Relation Compatible with Group Operation/CRG4

### R

- Reflexive Closure of Relation Compatible with Operation is Compatible
- Reflexive Reduction of Relation Compatible with Group Operation is Compatible
- Relation Compatible with Group Operation is Reflexive or Antireflexive
- Relation Compatible with Group Operation is Strongly Compatible
- Relations Compatible with Group Form Complete Boolean Algebra
- Relations Compatible with Operation Form Complete Distributive Lattice

### S

- Set Difference of Relations Compatible with Group Operation is Compatible
- Set Union Preserves Subsets
- Set Union Preserves Subsets/Corollary/Proof 1
- Subset Relation is Compatible with Subset Product
- Subset Relation is Compatible with Subset Product/Corollary 1
- Superset Relation is Compatible with Subset Product
- Symmetric Closure of Relation Compatible with Operation is Compatible