Independent axiom set

For an independent axiom set, semantic entailment of any one axiom from the remaining axioms fails. By the completeness theorem, this is equivalent to no member being formally derivable from the others in a complete calculus. To establish independence it suffices to give, for each axiom, a first-order structure satisfying all the other axioms and failing that one.

The usual associative, identity, two-sided inverse and commutativity axioms form an independent axiom set. A three-element commutative unital operation with self-inverses can fail associativity; a constant-zero operation can satisfy the other three axioms while failing the identity condition; natural-number addition fails the inverse condition; and a noncommutative group fails commutativity. Each example must be checked against all three retained axioms.