Axioms for fuzzy intersection

Statement

*Axiom i1. Boundary condition

i(a, 1) = a

*Axiom i2. Monotonicity

b ≤ d implies i(a, b) ≤ i(a, d)

*Axiom i3. Commutativity

i(a, b) = i(b, a)

*Axiom i4. Associativity

i(a, i(b, d)) = i(i(a, b), d)

*Axiom i5. Continuity

i is a continuous function

*Axiom i6. Subidempotency

i(a, a) < a for all 0 < a < 1

*Axiom i7. Strict monotonicity

i (a₁, b₁) < i (a₂, b₂) if a₁ < a₂ and b₁ < b₂

Axioms i1 up to i4 define a t-norm (a.k.a. fuzzy intersection). The standard t-norm min is the only idempotent t-norm (that is, i (a₁, a₁) = a for all a ∈ [0,1]).