Axioms for fuzzy complements

Statement

*Axiom c1. Boundary condition

c(0) = 1 and c(1) = 0

*Axiom c2. Monotonicity

For all a, b ∈ [0, 1], if a < b, then c(a) > c(b)

*Axiom c3. Continuity

c is continuous function.

*Axiom c4. Involutions

c is an involution, which means that c(c(a)) = a for each a ∈ [0,1]

c is a strong negator (a.k.a. fuzzy complement).

A function c satisfying axioms c1 and c3 has at least one fixpoint a^* with c(a^*) = a^*, and if axiom c2 is fulfilled as well there is exactly one such fixpoint. For the standard negator c(x) = 1-x the unique fixpoint is a^* = 0.5 .