Associative and quasitrivial operations on finite sets

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Associative and quasitrivial operations

on finite sets

Jean-Luc Marichal

University of Luxembourg Luxembourg

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Single-peaked orderings

Motivating example (Romero, 1978)

Suppose you are asked to order the following six objects in

decreasing preference:

a

1

:

0 sandwich

a

2

:

1 sandwich

a

3

:

2 sandwiches

a

4

:

3 sandwiches

a

5

:

4 sandwiches

a

6

:

more than 4 sandwiches

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Single-peaked orderings

a1: 0 sandwich a2: 1 sandwich a3: 2 sandwiches a4: 3 sandwiches a5: 4 sandwiches

a6: more than 4 sandwiches

after a good lunch: a1≺ a2≺ a3≺ a4≺ a5≺ a6 if you are starving: a6≺ a5≺ a4≺ a3≺ a2≺ a1

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Single-peaked orderings

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Single-peaked orderings

Definition (Black, 1948)

Let ≤ and  be total orderings on Xn= {a1, . . . , an}.

Then  is said to besingle-peaked w.r.t. ≤if for any ai, aj, ak ∈ Xn such that ai < aj < ak we have aj ≺ ai or aj ≺ ak.

Let us assume that Xn= {a1, . . . , an} is endowed with the ordering a1< · · · < an For n = 4 a1≺ a2≺ a3≺ a4 a4≺ a3≺ a2≺ a1 a2≺ a1≺ a3≺ a4 a3≺ a2≺ a1≺ a4 a2≺ a3≺ a1≺ a4 a3≺ a2≺ a4≺ a1 a2≺ a3≺ a4≺ a1 a3≺ a4≺ a2≺ a1

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Single-peaked orderings

Recall that aweak ordering(ortotal preordering) on Xn is a binary relation - on Xn that is total and transitive.

Defining a weak ordering on Xn amounts to defining an ordered partition of Xn

C1≺ · · · ≺ Ck

where C1, . . . , Ck are the equivalence classes defined by ∼ For n = 3, we have 13 weak orderings

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Single-peaked orderings

Definition

Let ≤ be a total ordering on Xnand let - be a weak ordering on Xn. We say that - is weakly single-peaked w.r.t. ≤ if for any ai, aj, ak ∈ Xn such that ai < aj < ak we have aj≺ ai or aj ≺ ak or ai ∼ aj ∼ ak.

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Connectedness and Contour plots

Let F : Xn2→ Xn be an operation on Xn= {1, . . . , n}

Definition

The points (u, v ) and (x , y ) of X2

n are said to beF -connectedif F (u, v ) = F (x , y )

The point (x , y ) of X2

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Connectedness and Contour plots

Definition

For any x ∈ Xn, theF -degree of x, denoteddegF(x ), is the number of points (u, v ) 6= (x , x ) such that F (u, v ) = F (x , x )

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Quasitriviality

Definition

F : Xn2→ Xn is said to be

quasitrivial(orconservative) if

F (x , y ) ∈ {x , y } (x , y ∈ Xn)

idempotentif

F (x , x ) = x (x ∈ Xn)

Fact. If F is quasitrivial, then it is idempotent

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Quasitriviality

Let ∆Xn = {(x , x ) | x ∈ Xn}

Fact

F : Xn2→ Xn is quasitrivial iff it is idempotent

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

r r r r r r r r r 1 2 3 r r r r r r r r r @ @ @ @ @ @ 1 2 3

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Neutral and annihilator elements

Definition

e ∈ Xn is said to be aneutral elementof F : Xn2→ Xnif F (x , e) = F (e, x ) = x , x ∈ Xn

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Neutral and annihilator elements

Proposition

Assume that F : Xn2→ Xnis quasitrivial.

e ∈ Xn is a neutral element of F iff degF(e) = 0

a ∈ Xnis an annihilator element of F iff degF(a) = 2n − 2.

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Associative, quasitrivial, and commutative operations

Theorem

Let F : X2

n → Xn. The following assertions are equivalent.

(i) F is associative, quasitrivial, and commutative (ii) F = max for some total ordering  on Xn The total ordering  is uniquely determined as follows:

x  y ⇐⇒ degF(x ) ≤ degF(y )

Fact. There are exactly n! such operations

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Associative, quasitrivial, and commutative operations

Theorem

Let F : X2

n → Xn. The following assertions are equivalent.

(i) F is associative, quasitrivial, and commutative (ii) F = maxfor some total ordering  on Xn

(iii) F is quasitrivial and {degF(x ) | x ∈ Xn} = {0, 2, 4, . . . , 2n − 2}

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Associative, quasitrivial, and commutative operations

Definition. F : X2

n → Xn is said to be ≤-preserving for some total ordering ≤ on Xn if for

any x , y , x0, y0∈ Xnsuch that x ≤ x0and y ≤ y0, we have F (x , y ) ≤ F (x0, y0)

Theorem

Let F : X2

n → Xn. The following assertions are equivalent.

(i) F is associative, quasitrivial, and commutative (ii) F = maxfor some total ordering  on Xn

(iii) F is quasitrivial and {degF(x ) | x ∈ Xn} = {0, 2, 4, . . . , 2n − 2}

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Associative, quasitrivial, and commutative operations

Definition.

Auninorm on Xn is an operation F : Xn2→ Xnthat has a neutral element e ∈ Xn

and is

associative commutative

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Associative, quasitrivial, and commutative operations

Theorem

Let F : X2

n → Xn. The following assertions are equivalent.

(i) F is associative, quasitrivial, and commutative (ii) F = maxfor some total ordering  on Xn

(iii) F is quasitrivial and {degF(x ) | x ∈ Xn} = {0, 2, 4, . . . , 2n − 2}

(iv) F is quasitrivial, commutative, and ≤-preserving for some total ordering ≤ on Xn

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Associative, quasitrivial, and commutative operations

Assume that Xn= {1, . . . , n} is endowed with the usual total ordering ≤n

defined by 1 <n· · · <nn Theorem

Let F : X2

n → Xn. The following assertions are equivalent.

(i) F is quasitrivial, commutative, and ≤n-preserving (⇒ associative)

(ii) F = maxfor some total ordering  on Xnthat is single-peaked w.r.t. ≤n

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Associative, quasitrivial, and commutative operations

Remark.

There are n! operations F : Xn2→ Xn that are associative, quasitrivial, and commutative.

There are 2n−1 of them that are ≤

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Associative and quasitrivial operations

Examples of noncommutative operations

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Associative and quasitrivial operations

Definition.

Theprojection operationsπ1: Xn2→ Xnand π2: Xn2→ Xn are respectively defined by

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Associative and quasitrivial operations

Assume that Xn= {1, . . . , n} is endowed with a weak ordering -Ordinal sum of projections

osp-: Xn2→ Xn -6 π1or π2 π1or π2 max max

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Associative and quasitrivial operations

Theorem (L¨anger 1980, Kepka 1981)

Let F : Xn2→ Xn. The following assertions are equivalent.

(i) F is associative and quasitrivial

(ii) F = osp- for some weak ordering - on Xn The weak ordering - is uniquely determined as follows:

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Associative and quasitrivial operations

How to check whether a quasitrivial operation F : Xn2→ Xnis associative?

1. Order the elements of Xn according to the weak ordering - defined by

x - y ⇐⇒ degF(x ) ≤ degF(y )

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Associative and quasitrivial operations

Which ones are ≤-preserving?

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Associative and quasitrivial operations

Assume that Xn= {1, . . . , n} is endowed with the usual total ordering ≤n defined by 1 <n· · · <nn

Theorem

Let F : Xn2→ Xn. The following assertions are equivalent.

(i) F is associative, quasitrivial, and ≤n-preserving

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Final remarks

1. We have introduced and identified a number of integer sequences in http://oeis.org

Number of associative and quasitrivial operations: A292932 Number of associative, quasitrivial, and ≤n-preserving operations: A293005

Number of weak orderings on Xnthat are weakly single-peaked w.r.t. ≤n: A048739

· · ·

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Some references

N. L. Ackerman.

A characterization of quasitrivial n-semigroups.

To appear in Algebra Universalis.

S. Berg and T. Perlinger.

Single-peaked compatible preference profiles: some combinatorial results.

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

T. Kepka.

Quasitrivial groupoids and balanced identities.

Acta Univ. Carolin. - Math. Phys., 22(2):49–64, 1981.

H. L¨anger.

The free algebra in the variety generated by quasi-trivial semigroups.

Semigroup forum, 20:151–156, 1980.

N. J. A. Sloane (editor).

The On-Line Encyclopedia of Integer Sequences.

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