Relation ASSOCIATIVE

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

Slots on this relation:

Instance-Of: Relation

Arity: 2

Equivalence Axioms:

(<=> (Associative ?Op ?Domain)
     (Forall (?X ?Y ?Z)
             (=> (Instance-Of ?X ?Domain)
                 (Instance-Of ?Y ?Domain)
                 (Instance-Of ?Z ?Domain)
                 (= (Value ?Op ?X (Value ?Op ?Y ?Z))
                    (Value ?Op (Value ?Op ?X ?Y) ?Z)))))

Other Related Axioms:

(<=> (Associative ?Op ?Domain)
     (Forall (?X ?Y ?Z)
             (=> (Instance-Of ?X ?Domain)
                 (Instance-Of ?Y ?Domain)
                 (Instance-Of ?Z ?Domain)
                 (= (Value ?Op ?X (Value ?Op ?Y ?Z))
                    (Value ?Op (Value ?Op ?X ?Y) ?Z)))))

(<=> (Semigroup ?Domain ?Op ?Id)
     (And (Binary-Operator-On ?Op ?Domain)
          (Associative ?Op ?Domain)
          (Identity-Element-For ?Id ?Op ?Domain)))

(<=> (Group ?Domain ?Op ?Id)
     (And (Binary-Operator-On ?Op ?Domain)
          (Associative ?Op ?Domain)
          (Identity-Element-For ?Id ?Op ?Domain)
          (Invertible ?Op ?Id ?Domain)))