Interactive visualization elaborated on the basis of: A. Pradera et al. (2016) DOI: 10.1016/j.ins.2015.09.033 ↗

Classification of Aggregation Functions and Generated Implication Properties

Multi-select filtering: click any number of properties on the filter bar or directly on the diagram to highlight multiple classes, inspect their intersection (∩), and verify compatibility.

🌐 Polski (PL)
Selected: 0 Click buttons to toggle filters. When ≥ 2 compatible options are selected, their exact geometric intersection (∩) is highlighted on the diagram.
Sample intersections:
4 Main Classes: 0 / 1 Elements:
Strictness & Divisors:
FULL DIAGRAM OVERVIEW

All classes and properties of aggregation functions (click multiple options to explore their intersection)

A : [0,1]² → [0,1], A(0,0)=0, A(1,1)=1
CONJUNCTIVE A(x, y) ≤ min(x, y) AVERAGING min(x, y) ≨ A(x, y) ≨ max(x, y) DISJUNCTIVE A(x, y) ≥ max(x, y) MIXED hybrid functions CONJUNCTORS AE(0): A(x,0) = A(0,x) = 0 DISJUNCTORS AE(1): A(x,1) = A(1,x) = 1 1Strict A(x,y)=1 ⇒ x=y=1 0Strict A(x,y)=0 ⇒ x=y=0 1Strict ∩ 0Strict 1Strict ∩ 0Strict Semi-copulas NE(1): A(x,1) = A(1,x) = x Dual semi-copulas NE(0): A(x,0) = A(0,x) = x 0Div A(a,b) = 0 NC A(x,N(x)) = 0 S-NC y≤N(x) ⇔ A=0 1Div A(a,b) = 1 EM A(x,N(x)) = 1 S-EM y≥N(x) ⇔ A=1 0Div ∩ 1Div

Inclusion Relations Between Classes (Table 3, Prop. 18–21)

Left branch (0-properties):
S-NC(N) ⊂ NC(N) ⊂ 0Div ⊂ ¬(0Strict)  and  NC(N) ⊂ AE(0) ⊂ ¬(0Strict)
Right branch (1-properties):
S-EM(N) ⊂ EM(N) ⊂ 1Div ⊂ ¬(1Strict)  and  EM(N) ⊂ AE(1) ⊂ ¬(1Strict)
Conjunctive & Disjunctive:
NE(1) ⊂ CONJ ⊂ (AE(0) ∩ 1Strict)  ·  NE(0) ⊂ DISJ ⊂ (AE(1) ∩ 0Strict)
Averaging (AVG):
AVG ⊂ ¬(0Div) ∩ ¬(1Div)  (averaging functions have neither zero nor one divisors)

Dictionary: Disjunctor A ⇄ Implication I_A(x,y) = A(1−x, y) (Table 6, Fig. 3)

A satisfies (DISJ) ⇔ I_A ≥ I_max (Kleene–Dienes)
A satisfies (AVG) ⇔ I_(D_avg)⊥ ≤ I_A ≨ I_max
A satisfies (EM) ⇔ (IP) I_A(x,x) = 1
A satisfies (S-EM) ⇔ (OP) x ≤ y ⇔ I_A(x,y) = 1
A does not satisfy (1Div) ⇔ (LT) I_A(x,y)=1 ⇔ (x=0 ∨ y=1)
A satisfies (0Strict) ⇔ (LF) I_A(x,y)=0 ⇔ (x=1 ∧ y=0)
A satisfies L-NE(0) ⇔ (NT) I_A(1,y) = y
(COMM) ⇔ (CP)  ·  (ASSOC) ⇔ LI(A^d)
Elaborated on the basis of the article

Diagram and property summary elaborated on the basis of: A. Pradera, G. Beliakov, H. Bustince, B. De Baets, "A review of the relationships between implication, negation and aggregation functions from the point of view of material implication", Information Sciences, vol. 329, pp. 357–380, 2016 (in particular Fig. 1, p. 365; Fig. 3, p. 371; Tables 1, 3, 6). DOI: 10.1016/j.ins.2015.09.033

Open Article (DOI) ↗