On quasitrivial and associative operations University of Zielona G´ora Jimmy Devillet

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On quasitrivial and associative operations

University of Zielona G´ora

Jimmy Devillet

in collaboration with Miguel Couceiro and Jean-Luc Marichal University of Luxembourg

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Connectedness and Contour Plots

Let X be a nonempty set and let F : X² → X Definition

The points (x , y ), (u, v ) ∈ X² are F-connected if F (x , y ) = F (u, v )

The point (x , y ) ∈ X² is F-isolatedif it is not F -connected to another point in X²

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Connectedness and Contour Plots

For any integer n ≥ 1, let Xn= {1, ..., n} endowed with ≤ Example. F (x , y ) = max{x , y } on (X4, ≤)

s s s s

- 6

1 2 3 4

1 2

3 4

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Quasitriviality and Idempotency

Definition

F : X² → X is said to be quasitrivialif

F (x , y ) ∈ {x , y } idempotent if

F (x , x ) = x

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Graphical interpretation of quasitriviality

Let ∆_X = {(x , x ) | x ∈ X } Proposition

F : X² → X is quasitrivial iff it is idempotent

every point (x , y ) /∈ ∆_X is F -connected to either (x , x ) or (y , y )

s s s

1 2

3

s s s

1

2 3

@

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Graphical interpretation of the neutral element

Definition. An element e ∈ X is said to be aneutral element of F : X² → X if

F (x , e) = F (e, x ) = x Proposition

Assume F : X²→ X is quasitrivial and let e ∈ X . Then e is a neutral element of F iff (e, e) is F -isolated

s s s

1 2

3

s s s

1

2 3

@

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Degree sequence

Recall that X_n= {1, ..., n}

Definition. Assume F : X_n² → X_n and let z ∈ X_n. TheF-degree of z, denoted deg_F(z), is the number of points (x , y ) 6= (z, z) such that F (x , y ) = F (z, z)

Definition. Assume F : X_n² → X_n. Thedegree sequence of F, denoted deg_F, is the nondecreasing n-element sequence of the F -degrees deg_F(x ), x ∈ Xn

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Degree sequence

s s s s

1 2

3 4

1 < 2 < 3 < 4

deg_F = (0, 2, 4, 6)

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Graphical interpretation of the annihilator

Definition. An element a ∈ X is said to be anannihilatorof F : X² → X if

F (x , a) = F (a, x ) = a

Proposition

Assume F : X_n²→ X_n is quasitrivial and let a ∈ X_n. Then a is an annihilator iff deg_F(a) = 2n − 2

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A class of associative operations

We are interested in the class of operations F : X²→ X that are associative

quasitrivial symmetric

Note : We will assume later that F is nondecreasing w.r.t. some total ordering on X

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A first characterization

Theorem (L¨anger, 1980)

F : X² → X is associative, quasitrivial and symmetric iff there exists a total ordering ≤ on X such that F = max≤.

s s s s

1 2

3 4

1 < 2 < 3 < 4

s s s s

2 4

3 1

2 ≺ 4 ≺ 3 ≺ 1

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A second characterization

Theorem

Let F : X² → X . If X = X_n then TFAE

(i) F is associative, quasitrivial and symmetric (ii) F = max≤ for some total ordering ≤ on Xn

(iii) F is quasitrivial and deg_F = (0, 2, 4, . . . , 2n − 2)

There are exactly n! operations F : X_n² → X_nsatifying any of the conditions (i)–(iii). Moreover, the total ordering ≤ considered in (ii) is determined by the condition: x y iff deg_F(x ) ≤ deg_F(y ).

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Operations on X

₃

r r r r r r r r r

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The nondecreasing case

s s s s

1 2

3 4

1 < 2 < 3 < 4

s s s s

2 4

3 1

2 ≺ 4 ≺ 3 ≺ 1

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Single-peaked total orderings

Definition.(Black, 1948) Let ≤, be total orderings on X . The total ordering is said to besingle-peaked w.r.t. ≤if for all a, b, c ∈ X such that a < b < c we have b ≺ a or b ≺ c Example. The total ordering on

X₄= {1 < 2 < 3 < 4}

defined by

3 ≺ 2 ≺ 4 ≺ 1 is single-peaked w.r.t. ≤

Note : There are exactly 2ⁿ⁻¹ single-peaked total orderings on (X_n, ≤).

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Single-peaked total orderings

s s s s

1 2

3 4

1 < 2 < 3 < 4

s s s s

3 2

4 1

3 ≺ 2 ≺ 4 ≺ 1

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A third characterization

Theorem

Let ≤ be a total ordering on X and let F : X² → X . TFAE (i) F est associative, quasitrivial, symmetric and nondecreasing (ii) F = max for some total ordering on X that is

single-peaked w.r.t. ≤

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A fourth characterization

Theorem

Let ≤ be a total ordering on X and let F : X² → X . If (X , ≤) = (X_n, ≤) then TFAE

(i) F is associative, quasitrivial, symmetric and nondecreasing (ii) F = max for some total ordering on Xn that is

single-peaked w.r.t. ≤

(iii) F is quasitrivial, nondecreasing and deg_F = (0, 2, 4, . . . , 2n − 2)

There are exactly 2ⁿ⁻¹ operations F : X_n² → X_n satisfying any of the conditions (i)–(iii).

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Operations on X

₃

s s s

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A more general class of associative operations

We are interested in the class of operations F : X²→ X that are associative

quasitrivial

Note : We will assume later that F is nondecreasing w.r.t. some total ordering on X

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Weak orderings

Recall that a binary relation R on X is said to be totalif ∀x , y : xRy or yRx

transitiveif ∀x , y , z: xRy and yRz implies xRz

Aweak ordering on X is a binary relation . on X that is total and transitive. We denote the symmetric and asymmetric parts of . by

∼ and <, respectively.

Recall that ∼ is an equivalence relation on X and that < induces a total ordering on the quotient set X / ∼

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A fifth characterization

Theorem (Mclean, 1954, Kimura, 1958)

F : X² → X is associative and quasitrivial iff there exists a weak ordering . on X such that

F |_A×B =

(max_.|_A×B, if A 6= B,

π₁|_A×B or π₂|_A×B, if A = B, ∀A, B ∈ X / ∼

s s s

1 2

3

1 < 2 < 3

s s s

2 1

3

2 ≺ 1 ∼ 3

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A fifth characterization

Theorem (Mclean, 1954, Kimura, 1958)

F : X² → X is associative and quasitrivial iff there exists a weak ordering . on X such that

F |A×B =

(max_.|_A×B, if A 6= B,

π₁|_A×B or π₂|_A×B, if A = B, ∀A, B ∈ X / ∼

Moreover, if X = Xn the weak ordering . is determined by the condition: x . y iff degF(x ) ≤ deg_F(y ).

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Operations on X

₃

q q q q q q q q q

q q q

q q q q q q

q q q q q q q q q

q q qq q qq q q

q q q q q q q q q

q q qq q qq q q

q q q q q q q q q

q q qq q qq q q

q q q q q q q q q

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The nondecreasing case

s s s

1 2

3

1 < 2 < 3

s s s

2 1

3

2 ≺ 1 ∼ 3

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Weakly single-peaked weak orderings

Definition. Let ≤ be a total ordering on X and let - be a weak ordering on X . The weak ordering - is said to beweakly

single-peaked w.r.t. ≤if for any a, b, c ∈ X such that a < b < c we have b ≺ a or b ≺ c or a ∼ b ∼ c

Example. The weak ordering - on

X₄= {1 < 2 < 3 < 4}

defined by

2 ≺ 1 ∼ 3 ≺ 4 is weakly single-peaked w.r.t. ≤

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Weakly single-peaked weak orderings

s s s s

1 2

3 4

1 < 2 < 3 < 4

s s s s

1 2

3 4

2 ≺ 1 ∼ 3 ≺ 4

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A sixth characterization

F |A×B =

(max_-|_A×B, if A 6= B,

π1|_A×B or π2|_A×B, if A = B, ∀A, B ∈ X / ∼ (∗)

Theorem

Let ≤ be a total ordering on X . F : X² → X is associative, quasitrivial, and nondecreasing w.r.t. ≤ iff F is of the form (∗) for some weak ordering - on X that is weakly single-peaked w.r.t. ≤

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Enumeration of associative and quasitrivial operations

Recall that if the generating function (GF) or the exponential generating function (EGF) of a given sequence (s_n)_n≥0 exist, then they are respectively defined as the power series

S (z) = X

n≥0

s_nzⁿ and S (z) =ˆ X

n≥0

s_nzⁿ n! .

Recall also that for any integers 0 ≤ k ≤ n theStirling number of the second kind_n

k is defined as

n k

= 1

k!

k

X

i =0

(−1)^k−ik i

iⁿ.

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Enumeration of associative and quasitrivial operations

For any integer n ≥ 1, let q(n) denote the number of associative and quasitrivial operations F : X_n² → X_n (OEIS : A292932) Theorem

For any integer n ≥ 0, we have the closed-form expression

q(n) =

n

X

i =0

2ⁱ

n−i

X

k=0

(−1)^kn k

n − k i

(i + k)! , n ≥ 0.

Moreover, its EGF is given by ˆQ(z) = 1/(z + 3 − 2e^z).

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Enumeration of associative and quasitrivial operations

In arXiv:1709.09162 we found also explicit formulas for

qe(n) : number of associative and quasitrivial operations F : X_n²→ X_n that have a neutral element (OEIS : A292933) qa(n) : number of associative and quasitrivial operations F : X_n²→ X_n that have an annihilator (OEIS : A292933) qea(n) : number of associative and quasitrivial operations F : X_n²→ X_n that have a neutral element and an annihilator (OEIS : A292934)

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Enumeration of associative quasitrivial and nondecreasing operations

For any integer n ≥ 0 we denote by v (n) the number of associative, quasitrivial, and nondecreasing operations F : X_n² → X_n (OEIS : A293005)

Theorem

For any integer n ≥ 0, we have the closed-form expression 3 v (n) + 2 = P

k≥03^k(2 _2kⁿ + 3 _2k+1ⁿ ) , n ≥ 0.

Moreover, its GF is given by V (z) = z(z + 1)/(2z³− 3z + 1).

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Enumeration of associative quasitrivial and nondecreasing operations

In arXiv:1709.09162 we found also explicit formulas for

v_e(n) : number of associative, quasitrivial and nondecreasing operations F : X_n²→ X_n that have a neutral element (OEIS : A002605)

va(n) : number of associative, quasitrivial and nondecreasing operations F : X_n²→ X_n that have an annihilator (OEIS : A293006)

v_ea(n) : number of associative, quasitrivial and nondecreasing operations F : X_n²→ X_n that have a neutral element and an annihilator (OEIS : A293007)

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Selected references

D. Black. On the rationale of group decision-making. J Polit Economy, 56(1):23–34, 1948

M. Couceiro, J. Devillet, and J.-L. Marichal. Characterizations of idempotent discrete uninorms. Fuzzy Sets and Systems. In press.

https://doi.org/10.1016/j.fss.2017.06.013

M. Couceiro, J. Devillet, and J.-L. Marichal. Quasitrivial semigroups:

characterizations and enumerations. Submitted for publication. arXiv:1709.09162.

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. Submitted for publication.

arXiv:1705.00719.

N. Kimura. The structure of idempotent semigroups. I. Pacific J. Math., 8:257–275, 1958.

H. L¨anger. The free algebra in the variety generated by quasi-trivial semigroups.

Semigroup forum, 20:151–156, 1980.

D. McLean. Idempotent semigroups. Amer. Math. Monthly, 61:110–113, 1954.