Axioms for fuzzy union

Statement

*Axiom u1. Boundary condition

u(a, 0) =u(0 ,a) = a

*Axiom u2. Monotonicity

b ≤ d implies u(a, b) ≤ u(a, d)

*Axiom u3. Commutativity

u(a, b) = u(b, a)

*Axiom u4. Associativity

u(a, u(b, d)) = u(u(a, b), d)

*Axiom u5. Continuity

u is a continuous function

*Axiom u6. Superidempotency

u(a, a) > a for all 0 < a < 1

*Axiom u7. Strict monotonicity

a₁ < a₂ and b₁ < b₂ implies u(a₁, b₁) < u(a₂, b₂)

Axioms u1 up to u4 define a t-conorm (a.k.a. s-norm or fuzzy union). The standard t-conorm max is the only idempotent t-conorm (i. e. u (a1, a1) = a for all a ∈ [0,1]).