Relation IRREFLEXIVE

• Defined in theory: Abstract-algebra
• Source code: abstract-algebra.lisp

Slots on this relation:

Instance-Of: Relation

Arity: 2

Domain: Binary-relation

Equivalence Axioms:

(<=> (Irreflexive ?Rel ?Domain)
     (And (Binary-Relation ?Rel)
          (Forall (?X)
                  (=> (Instance-Of ?X ?Domain)
                      (Not (Holds ?Rel ?X ?X))))))

Other Related Axioms:

(=> (Irreflexive $X $Y) (Binary-Relation $X))

(<=> (Irreflexive ?Rel ?Domain)
     (And (Binary-Relation ?Rel)
          (Forall (?X)
                  (=> (Instance-Of ?X ?Domain)
                      (Not (Holds ?Rel ?X ?X))))))

(<=> (Partial-Order ?Domain ?Rel)
     (And (Irreflexive ?Rel ?Domain)
          (Asymmetric ?Rel ?Domain)
          (Transitive ?Rel ?Domain)))

(<=> (Linear-Order ?Domain ?Rel)
     (And (Irreflexive ?Rel ?Domain)
          (Asymmetric ?Rel ?Domain)
          (Transitive ?Rel ?Domain)
          (Trichotomizes ?Rel ?Domain)))