On idempotent n-ary uninorms

On idempotent n-ary uninorms

Jimmy Devillet

University of Luxembourg Luxembourg

in collaboraton with Gergely Kiss and Jean-Luc Marichal

Part I: Ultrabisymmetry

Associativity and symmetry

Definition.

F : X ³ → X is said to be

associative if for all x ₁ , x ₂ , x ₃ , x ₄ , x ₅ ∈ X F (F (x 1 , x 2 , x 3 ), x 4 , x 5 )

= F (x ₁ , F (x ₂ , x ₃ , x ₄ ), x ₅ )

= F (x ₁ , x ₂ , F (x ₃ , x ₄ , x ₅ ))

symmetric if F (x 1 , x 2 , x 3 ) is invariant under any permutation of x ₁ , x ₂ , x ₃

Example. F (x, y, z ) = x + y + z on X = R

Associativity and symmetry

Definition.

F : X ³ → X is said to be

associative if for all x ₁ , x ₂ , x ₃ , x ₄ , x ₅ ∈ X F (F (x 1 , x 2 , x 3 ), x 4 , x 5 )

= F (x ₁ , F (x ₂ , x ₃ , x ₄ ), x ₅ )

= F (x ₁ , x ₂ , F (x ₃ , x ₄ , x ₅ ))

symmetric if F (x 1 , x 2 , x 3 ) is invariant under any permutation of x ₁ , x ₂ , x ₃

Example. F (x, y, z ) = x + y + z on X = R

Associativity and symmetry

Definition.

F : X ³ → X is said to be

associative if for all x ₁ , x ₂ , x ₃ , x ₄ , x ₅ ∈ X F (F (x 1 , x 2 , x 3 ), x 4 , x 5 )

= F (x ₁ , F (x ₂ , x ₃ , x ₄ ), x ₅ )

= F (x ₁ , x ₂ , F (x ₃ , x ₄ , x ₅ ))

symmetric if F (x 1 , x 2 , x 3 ) is invariant under any permutation of x ₁ , x ₂ , x ₃

Example. F (x, y, z ) = x + y + z on X = R

Associativity and symmetry

Definition.

F : X ³ → X is said to be

associative if for all x ₁ , x ₂ , x ₃ , x ₄ , x ₅ ∈ X F (F (x 1 , x 2 , x 3 ), x 4 , x 5 )

= F (x ₁ , F (x ₂ , x ₃ , x ₄ ), x ₅ )

= F (x ₁ , x ₂ , F (x ₃ , x ₄ , x ₅ ))

symmetric if F (x 1 , x 2 , x 3 ) is invariant under any permutation of x ₁ , x ₂ , x ₃

Example. F (x, y, z ) = x + y + z on X = R

Bisymmetry

Definition. (Acz´ el, 1946)

F : X ³ → X is said to be bisymmetric if

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

for all





x ₁₁ x ₁₂ x ₁₃ x ₂₁ x ₂₂ x ₂₃ x 31 x 32 x 33



 ∈ X ^3×3

Example. F (x, y, z ) = x + y + z on X = R

Bisymmetry

Definition. (Acz´ el, 1946)

F : X ³ → X is said to be bisymmetric if

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

for all





x ₁₁ x ₁₂ x ₁₃ x ₂₁ x ₂₂ x ₂₃ x 31 x 32 x 33



 ∈ X ^3×3

Example. F (x, y, z ) = x + y + z on X = R

Ultrabisymmetry

Definition.

We say that F : X ³ → X is ultrabisymmetric if

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 )) is invariant when replacing x ij by x kl for all i , j , k, l ∈ {1, 2, 3}





x 11 x 12 x 13

x ₂₁ x ₂₂ x ₂₃ x ₃₁ x ₃₂ x ₃₃



 →





x 11 x 12 x 13

x ₃₃ x ₂₂ x ₂₃ x ₃₁ x ₃₂ x ₂₁



 ∈ X ^3×3

Example. F (x, y, z ) = x + y + z on X = R

Fact. ultrabisymmetry = ⇒ bisymmetry

Ultrabisymmetry

Definition.

We say that F : X ³ → X is ultrabisymmetric if

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 )) is invariant when replacing x ij by x kl for all i , j , k, l ∈ {1, 2, 3}





x 11 x 12 x 13

x ₂₁ x ₂₂ x ₂₃ x ₃₁ x ₃₂ x ₃₃



 →





x 11 x 12 x 13

x ₃₃ x ₂₂ x ₂₃ x ₃₁ x ₃₂ x ₂₁



 ∈ X ^3×3

Example. F (x, y, z ) = x + y + z on X = R

Fact. ultrabisymmetry = ⇒ bisymmetry

Ultrabisymmetry

Definition.

We say that F : X ³ → X is ultrabisymmetric if

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 )) is invariant when replacing x ij by x kl for all i , j , k, l ∈ {1, 2, 3}





x 11 x 12 x 13

x ₂₁ x ₂₂ x ₂₃ x ₃₁ x ₃₂ x ₃₃



 →





x 11 x 12 x 13

x ₃₃ x ₂₂ x ₂₃ x ₃₁ x ₃₂ x ₂₁



 ∈ X ^3×3

Example. F (x, y, z ) = x + y + z on X = R

Fact. ultrabisymmetry = ⇒ bisymmetry

Ultrabisymmetry

Definition.

We say that F : X ³ → X is ultrabisymmetric if

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 )) is invariant when replacing x ij by x kl for all i , j , k, l ∈ {1, 2, 3}





x 11 x 12 x 13

x ₂₁ x ₂₂ x ₂₃ x ₃₁ x ₃₂ x ₃₃



 →





x 11 x 12 x 13

x ₃₃ x ₂₂ x ₂₃ x ₃₁ x ₃₂ x ₂₁



 ∈ X ^3×3

Example. F (x, y, z ) = x + y + z on X = R

Fact. ultrabisymmetry = ⇒ bisymmetry

Associativity and bisymmetry

F (x, y, z ) = x + y + z is associative

symmetric

⇒ bisymmetric

⇒ ultrabisymmetric

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= (x ₁₁ + x ₁₂ + x ₁₃ ) + (x ₂₁ + x ₂₂ + x ₂₃ ) + (x ₃₁ + x ₃₂ + x ₃₃ )

= x ₁₁ + x ₁₂ + x ₁₃ + x ₂₁ + x ₂₂ + x ₂₃ + x ₃₁ + x ₃₂ + x ₃₃

= x ₁₁ + x ₂₁ + x ₃₁ + x ₁₂ + x ₂₂ + x ₃₂ + x ₁₃ + x ₂₃ + x ₃₃

= (x 11 + x 21 + x 31 ) + (x 12 + x 22 + x 32 ) + (x 13 + x 23 + x 33 )

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

Associativity and bisymmetry

F (x, y, z ) = x + y + z is associative

symmetric

⇒ bisymmetric

⇒ ultrabisymmetric

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= (x ₁₁ + x ₁₂ + x ₁₃ ) + (x ₂₁ + x ₂₂ + x ₂₃ ) + (x ₃₁ + x ₃₂ + x ₃₃ )

= x ₁₁ + x ₁₂ + x ₁₃ + x ₂₁ + x ₂₂ + x ₂₃ + x ₃₁ + x ₃₂ + x ₃₃

= x ₁₁ + x ₂₁ + x ₃₁ + x ₁₂ + x ₂₂ + x ₃₂ + x ₁₃ + x ₂₃ + x ₃₃

= (x 11 + x 21 + x 31 ) + (x 12 + x 22 + x 32 ) + (x 13 + x 23 + x 33 )

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

Associativity and bisymmetry

F (x, y, z ) = x + y + z is associative

symmetric

⇒ bisymmetric

⇒ ultrabisymmetric

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= (x ₁₁ + x ₁₂ + x ₁₃ ) + (x ₂₁ + x ₂₂ + x ₂₃ ) + (x ₃₁ + x ₃₂ + x ₃₃ )

= x ₁₁ + x ₁₂ + x ₁₃ + x ₂₁ + x ₂₂ + x ₂₃ + x ₃₁ + x ₃₂ + x ₃₃

= x ₁₁ + x ₂₁ + x ₃₁ + x ₁₂ + x ₂₂ + x ₃₂ + x ₁₃ + x ₂₃ + x ₃₃

= (x 11 + x 21 + x 31 ) + (x 12 + x 22 + x 32 ) + (x 13 + x 23 + x 33 )

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

Associativity and bisymmetry

F (x, y, z ) = x + y + z is associative

symmetric

⇒ bisymmetric

⇒ ultrabisymmetric

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= (x ₁₁ + x ₁₂ + x ₁₃ ) + (x ₂₁ + x ₂₂ + x ₂₃ ) + (x ₃₁ + x ₃₂ + x ₃₃ )

= x ₁₁ + x ₁₂ + x ₁₃ + x ₂₁ + x ₂₂ + x ₂₃ + x ₃₁ + x ₃₂ + x ₃₃

= x ₁₁ + x ₂₁ + x ₃₁ + x ₁₂ + x ₂₂ + x ₃₂ + x ₁₃ + x ₂₃ + x ₃₃

= (x 11 + x 21 + x 31 ) + (x 12 + x 22 + x 32 ) + (x 13 + x 23 + x 33 )

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

Associativity and bisymmetry

F (x, y, z ) = x + y + z is associative

symmetric

⇒ bisymmetric

⇒ ultrabisymmetric

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= (x ₁₁ + x ₁₂ + x ₁₃ ) + (x ₂₁ + x ₂₂ + x ₂₃ ) + (x ₃₁ + x ₃₂ + x ₃₃ )

= x ₁₁ + x ₁₂ + x ₁₃ + x ₂₁ + x ₂₂ + x ₂₃ + x ₃₁ + x ₃₂ + x ₃₃

= x ₁₁ + x ₂₁ + x ₃₁ + x ₁₂ + x ₂₂ + x ₃₂ + x ₁₃ + x ₂₃ + x ₃₃

= (x 11 + x 21 + x 31 ) + (x 12 + x 22 + x 32 ) + (x 13 + x 23 + x 33 )

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

Associativity and bisymmetry

F (x, y, z ) = x + y + z is associative

symmetric

⇒ bisymmetric

⇒ ultrabisymmetric

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= (x ₁₁ + x ₁₂ + x ₁₃ ) + (x ₂₁ + x ₂₂ + x ₂₃ ) + (x ₃₁ + x ₃₂ + x ₃₃ )

= x ₁₁ + x ₁₂ + x ₁₃ + x ₂₁ + x ₂₂ + x ₂₃ + x ₃₁ + x ₃₂ + x ₃₃

= x ₁₁ + x ₂₁ + x ₃₁ + x ₁₂ + x ₂₂ + x ₃₂ + x ₁₃ + x ₂₃ + x ₃₃

= (x 11 + x 21 + x 31 ) + (x 12 + x 22 + x 32 ) + (x 13 + x 23 + x 33 )

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

Associativity and bisymmetry

F (x, y, z ) = x + y + z is associative

symmetric

⇒ bisymmetric

⇒ ultrabisymmetric

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= (x ₁₁ + x ₁₂ + x ₁₃ ) + (x ₂₁ + x ₂₂ + x ₂₃ ) + (x ₃₁ + x ₃₂ + x ₃₃ )

= x ₁₁ + x ₁₂ + x ₁₃ + x ₂₁ + x ₂₂ + x ₂₃ + x ₃₁ + x ₃₂ + x ₃₃

= x ₁₁ + x ₂₁ + x ₃₁ + x ₁₂ + x ₂₂ + x ₃₂ + x ₁₃ + x ₂₃ + x ₃₃

= (x 11 + x 21 + x 31 ) + (x 12 + x 22 + x 32 ) + (x 13 + x 23 + x 33 )

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

Associativity and bisymmetry

F (x, y, z ) = x + y + z is associative

symmetric

⇒ bisymmetric

⇒ ultrabisymmetric

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= (x ₁₁ + x ₁₂ + x ₁₃ ) + (x ₂₁ + x ₂₂ + x ₂₃ ) + (x ₃₁ + x ₃₂ + x ₃₃ )

= x ₁₁ + x ₁₂ + x ₁₃ + x ₂₁ + x ₂₂ + x ₂₃ + x ₃₁ + x ₃₂ + x ₃₃

= x ₁₁ + x ₂₁ + x ₃₁ + x ₁₂ + x ₂₂ + x ₃₂ + x ₁₃ + x ₂₃ + x ₃₃

= (x 11 + x 21 + x 31 ) + (x 12 + x 22 + x 32 ) + (x 13 + x 23 + x 33 )

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

Associativity and bisymmetry

F (x, y, z ) = x + y + z is associative

symmetric

⇒ bisymmetric

⇒ ultrabisymmetric

F (F (x 11 , x 12 , x 13 ), F (x 21 , x 22 , x 23 ), F (x 31 , x 32 , x 33 ))

= (x ₁₁ + x ₁₂ + x ₁₃ ) + (x ₂₁ + x ₂₂ + x ₂₃ ) + (x ₃₁ + x ₃₂ + x ₃₃ )

= x ₁₁ + x ₁₂ + x ₁₃ + x ₂₁ + x ₂₂ + x ₂₃ + x ₃₁ + x ₃₂ + x ₃₃

= x ₁₁ + x ₂₁ + x ₃₁ + x ₁₂ + x ₂₂ + x ₃₂ + x ₁₃ + x ₂₃ + x ₃₃

= (x 11 + x 21 + x 31 ) + (x 12 + x 22 + x 32 ) + (x 13 + x 23 + x 33 )

= F (F (x 11 , x 21 , x 31 ), F (x 12 , x 22 , x 32 ), F (x 13 , x 23 , x 33 ))

Associativity and bisymmetry

Proposition

associativity + symmetry = ⇒ ultrabisymmetry

= ⇒ bisymmetry

bisymmetry + symmetry = ⇒ ultrabisymmetry

Associativity and bisymmetry

Proposition

associativity + symmetry = ⇒ ultrabisymmetry

= ⇒ bisymmetry

bisymmetry + symmetry = ⇒ ultrabisymmetry

Quasitriviality

Definition

F : X ³ → X is said to be

quasitrivial (or conservative) if

F (x, y , z ) ∈ {x, y , z } (x, y, z ∈ X )

idempotent if

F (x, x, x) = x (x ∈ X )

Fact. quasitriviality = ⇒ idempotency

Quasitriviality

Definition

F : X ³ → X is said to be

quasitrivial (or conservative) if

F (x, y , z ) ∈ {x, y , z } (x, y, z ∈ X )

idempotent if

F (x, x, x) = x (x ∈ X )

Fact. quasitriviality = ⇒ idempotency

Quasitriviality

Definition

F : X ³ → X is said to be

quasitrivial (or conservative) if

F (x, y , z ) ∈ {x, y , z } (x, y, z ∈ X )

idempotent if

F (x, x, x) = x (x ∈ X )

Fact. quasitriviality = ⇒ idempotency

Quasitriviality

Definition

F : X ³ → X is said to be

quasitrivial (or conservative) if

F (x, y , z ) ∈ {x, y , z } (x, y, z ∈ X )

idempotent if

F (x, x, x) = x (x ∈ X )

Fact. quasitriviality = ⇒ idempotency

Associativity and bisymmetry

bisymmetry + symmetry = ⇒ ultrabisymmetry

Proposition

quasitriviality + ultrabisymmetry = ⇒ associativity + symmetry

= ⇒ bisymmetry

Corollary

quasitriviality + symmetry

⇓

associativity ⇐⇒ bisymmetry

Associativity and bisymmetry

bisymmetry + symmetry = ⇒ ultrabisymmetry

Proposition

quasitriviality + ultrabisymmetry = ⇒ associativity + symmetry

= ⇒ bisymmetry

Corollary

quasitriviality + symmetry

⇓

associativity ⇐⇒ bisymmetry

Associativity and bisymmetry

bisymmetry + symmetry = ⇒ ultrabisymmetry

Proposition

quasitriviality + ultrabisymmetry = ⇒ associativity + symmetry

= ⇒ bisymmetry

Corollary

quasitriviality + symmetry

⇓

associativity ⇐⇒ bisymmetry

Part II: Idempotent n-ary uninorms

Uninorm

Definition

e ∈ X is said to be a neutral element of F : X

→ X if

F(x, e, e) = F(e, x , e) = F (e, e, x ) = x x ∈ X

Definition. (Kiss et al., 2018)

A ternary uninorm on (X , ≤) is an operation F : X

→ X that has a neutral element e ∈ X

and is

associative symmetric

≤-nondecreasing

First characterization

Proposition

F : X

→ X is an idempotent ternary uninorm if and only if there exists an idempotent binary uninorm U : X

→ X such that

F (x , y , z) = U(min(x , y , z), max(x , y , z)) x , y , z ∈ X .

Single-peaked orderings

Definition. (Black, 1948)

Let ≤ and be total orderings on X .

Then is said to be single-peaked for ≤ if for all a, b, c ∈ X a < b < c = ⇒ b ≺ a or b ≺ c

Example. On X = {1, 2, 3, 4} consider ≤ and defined by 1 < 2 < 3 < 4 and 2 ≺ 3 ≺ 1 ≺ 4

- 6

1 2 3 4 4

1 3 2

@ @ A

A A r

r r

r

Single-peaked orderings

Definition. (Black, 1948)

Let ≤ and be total orderings on X .

Then is said to be single-peaked for ≤ if for all a, b, c ∈ X a < b < c = ⇒ b ≺ a or b ≺ c

Example. On X = {1, 2, 3, 4} consider ≤ and defined by 1 < 2 < 3 < 4 and 2 ≺ 3 ≺ 1 ≺ 4

- 6

1 2 3 4

4 1 3 2

@ @ A

A A r

r r

r

Alternative characterization

Theorem

Let F : X

→ X be an operation. The following assertions are equivalent.

(i) F is associative, quasitrivial, symmetric, and ≤-nondecreasing.

(ii) F is bisymmetric, quasitrivial, symmetric, and ≤-nondecreasing.

(iii) F = max for some total ordering on X that is single-peaked for ≤ If F has a neutral element, then (i)–(iii) are equivalent to

(iv) F is an idempotent ternary uninorm.

Example

- 6

1 2 3 4

1 2 3 4

< < <

r r r r r r r r r r r r r r r r

1 2

3 4

- 6

2 3 1 4

2 3 1 4

≺ ≺ ≺ r r r r r r r r r r r r r r r r

2 3

1 4

- 6

1 2 3 4

4 1 3 2

@ @ A

A A r

r r

r

Some references

S. Berg and T. Perlinger.

Single-peaked compatible preference profiles: some combinatorial results.

em Social Choice and Welfare 27(1):89–102, 2006.

D. Black.

On the rationale of group decision-making.

J Polit Economy, 56(1):23–34, 1948

B. De Baets, J. Fodor, D. Ruiz-Aguilera, and J. Torrens.

Idempotent uninorms on finite ordinal scales.

Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 17(1):1–14, 2009.

J. Devillet, G. Kiss, and J.-L. Marichal.

Characterizations of quasitrivial symmetric nondecreasing associative operations.

Semigroup Forum (2019) 98(1):154–171.

https://doi.org/10.1007/s00233-018-9980-z.

R. R. Yager and A. Rybalov.

Uninorm aggregation operators.

Fuzzy Sets and Systems, 80:111–120, 1996.