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 (aka fuzzy intersection). The standard t-norm min is the only idempotent t-norm (that is, i (a1, a1) = a for all a ∈ [0,1]). Fuzzy unions The union of two fuzzy sets A and B is specified in general by a binary operation on the unit interval function of the form u:[0,1]×[0,1] → [0,1]. For all x ∈ U: μA ∪ B(x) = u[μA(x), μB(x)]. Axioms for fuzzy union 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. Continuityu is a continuous function Axiom u6. Superidempotencyu(a, a) > a for all 0 < a < 1 Axiom u7. Strict monotonicitya1 < a2 and b1 < b2 implies u(a1, b1) < u(a2, b2) Axioms u1 up to u4 define a t-conorm (aka 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]). Aggregation operations Aggregation operations on fuzzy sets are operations by which several fuzzy sets are combined in a desirable way to produce a single fuzzy set. Aggregation operation on n fuzzy set (2 ≤ n) is defined by a functionh:[0,1]n → [0,1] Axioms for aggregation operations fuzzy sets Axiom h1. Boundary conditionh(0, 0, ..., 0) = 0 and h(1, 1, ..., 1) = one Axiom h2. MonotonicityFor any pair <a1, a2, ..., an> and <b1, b2, ..., bn> of n-tuples such that ai, bi ∈ [0,1] for all i ∈ Nn, if ai ≤ bi for all i ∈ Nn, then h(a1, a2, ...,an) ≤ h(b1, b2, ..., bn); that is, h is monotonic increasing in all its arguments. Axiom h3. Continuityh'' is a continuous function. See also Fuzzy logic Fuzzy set T-norm Type-2 fuzzy sets and systems De Morgan algebra Further reading References L.A. Zadeh. Fuzzy