Characterizations and classifications of quasitrivial ...€¦ · Characterizations and classi cations of quasitrivial semigroups Jimmy Devillet University of Luxembourg Luxembourg

Characterizations and classifications ofquasitrivial semigroups

Jimmy Devillet

University of LuxembourgLuxembourg

in collaboraton with Jean-Luc Marichal and Bruno Teheux

Part I: Single-plateauedness and 2-quasilinearity

Weak orderings

Recall that a weak ordering (or total preordering) on a set X is a binaryrelation - on X that is total and transitive.

Defining a weak ordering on X amounts to defining an ordered partitionof X

For X = {a1, a2, a3}, we have 13 weak orderings

a1 ≺ a2 ≺ a3 a1 ∼ a2 ≺ a3 a1 ∼ a2 ∼ a3

a1 ≺ a3 ≺ a2 a1 ≺ a2 ∼ a3

a2 ≺ a1 ≺ a3 a2 ≺ a1 ∼ a3

a2 ≺ a3 ≺ a1 a3 ≺ a1 ∼ a2

a3 ≺ a1 ≺ a2 a1 ∼ a3 ≺ a2

a3 ≺ a2 ≺ a1 a2 ∼ a3 ≺ a1

Weak orderings




a1 ≺ a2 ≺ a3 a1 ∼ a2 ≺ a3 a1 ∼ a2 ∼ a3

a1 ≺ a3 ≺ a2 a1 ≺ a2 ∼ a3

a2 ≺ a1 ≺ a3 a2 ≺ a1 ∼ a3

a2 ≺ a3 ≺ a1 a3 ≺ a1 ∼ a2

a3 ≺ a1 ≺ a2 a1 ∼ a3 ≺ a2

a3 ≺ a2 ≺ a1 a2 ∼ a3 ≺ a1

Weak orderings




a1 ≺ a2 ≺ a3 a1 ∼ a2 ≺ a3 a1 ∼ a2 ∼ a3

a1 ≺ a3 ≺ a2 a1 ≺ a2 ∼ a3

a2 ≺ a1 ≺ a3 a2 ≺ a1 ∼ a3

a2 ≺ a3 ≺ a1 a3 ≺ a1 ∼ a2

a3 ≺ a1 ≺ a2 a1 ∼ a3 ≺ a2

a3 ≺ a2 ≺ a1 a2 ∼ a3 ≺ a1

Single-plateaued weak orderings

Definition. (Black, 1948)Let ≤ be a total ordering on X and let - be a weak ordering on X .Then - is said to be single-plateaued for ≤ if

ai < aj < ak =⇒ aj ≺ ai or aj ≺ ak or ai ∼ aj ∼ ak

Examples. On X = {a1 < a2 < a3 < a4 < a5 < a6}

a3 ∼ a4 ≺ a2 ≺ a1 ∼ a5 ≺ a6 a3 ∼ a4 ≺ a2 ∼ a1 ≺ a5 ≺ a6

-

6

a1 a2 a3 a4 a5 a6

a6

a1 ∼ a5

a2

a3 ∼ a4

�� A

AA@@

r rr rr r

-

6

a1 a2 a3 a4 a5 a6

a6

a5

a1 ∼ a2

a3 ∼ a4

�� AAA@@

r r r r r r


Definition. (Black, 1948)Let ≤ be a total ordering on X and let - be a weak ordering on X .Then - is said to be single-plateaued for ≤ if

ai < aj < ak =⇒ aj ≺ ai or aj ≺ ak or ai ∼ aj ∼ ak

Examples. On X = {a1 < a2 < a3 < a4 < a5 < a6}

a3 ∼ a4 ≺ a2 ≺ a1 ∼ a5 ≺ a6 a3 ∼ a4 ≺ a2 ∼ a1 ≺ a5 ≺ a6

-

6

a1 a2 a3 a4 a5 a6

a6

a1 ∼ a5

a2

a3 ∼ a4

�� A

AA@@

r rr rr r

-

6

a1 a2 a3 a4 a5 a6

a6

a5

a1 ∼ a2

a3 ∼ a4

�� AAA@@

r r r r r r


Q: Given - is it possible to find ≤ for which - is single-plateaued?

Example: On X = {a1, a2, a3, a4} consider - and -′ defined by

a1∼a2≺a3∼a4 and a1≺′a2∼′a3∼′a4

Yes! Consider ≤ defined by a3 < a1 < a2 < a4

-

6

a3 a1 a2 a4

a3 ∼ a4

a1 ∼ a2

J

JJrr rr

No!




a1∼a2≺a3∼a4 and a1≺′a2∼′a3∼′a4


-

6

a3 a1 a2 a4

a3 ∼ a4

a1 ∼ a2

J

JJrr rr

No!




a1∼a2≺a3∼a4 and a1≺′a2∼′a3∼′a4


-

6

a3 a1 a2 a4

a3 ∼ a4

a1 ∼ a2

J

JJrr rr

No!




a1∼a2≺a3∼a4 and a1≺′a2∼′a3∼′a4


-

6

a3 a1 a2 a4

a3 ∼ a4

a1 ∼ a2

J

JJrr rr

No!




a1∼a2≺a3∼a4 and a1≺′a2∼′a3∼′a4


-

6

a3 a1 a2 a4

a3 ∼ a4

a1 ∼ a2

J

JJrr rr

No!

2-quasilinear weak orderings

Definition.We say that - is 2-quasilinear if

a ≺ b ∼ c ∼ d =⇒ a, b, c , d are not pairwise distinct

Proposition

Assume the axiom of choice.

- is 2-quasilinear ⇐⇒ ∃ ≤ for which - is single-plateaued

2-quasilinear weak orderings

Definition.We say that - is 2-quasilinear if

a ≺ b ∼ c ∼ d =⇒ a, b, c , d are not pairwise distinct

Proposition


- is 2-quasilinear ⇐⇒ ∃ ≤ for which - is single-plateaued

Part II: Quasitrivial semigroups

Quasitriviality

Definition

F : X 2 → X is said to be quasitrivial (or conservative) if

F (x , y) ∈ {x , y} x , y ∈ X

Example. F = max≤ on X = {1, 2, 3} endowed with the usual ≤

r r rr r rr r r

-

6

1 2 3

1

2

3

1

2

3

Quasitriviality

Definition


F (x , y) ∈ {x , y} x , y ∈ X


r r rr r rr r r

-

6

1 2 3

1

2

3

1

2

3

Quasitriviality

Definition


F (x , y) ∈ {x , y} x , y ∈ X


r r rr r rr r r

-

6

1 2 3

1

2

3

1

2

3

Projections

Definition.

The projection operations π1 : X 2 → X and π2 : X 2 → X are respectivelydefined by

π1(x , y) = x , x , y ∈ X

π2(x , y) = y , x , y ∈ X

Quasitrivial semigroups

Theorem (Langer, 1980)

F is associative and quasitrivial

m

∃ - : F |A×B =

{max- |A×B , if A 6= B,

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

-

6

π1 orπ2

π1 orπ2

max-

max-


Theorem (Langer, 1980)

F is associative and quasitrivial

m

∃ - : F |A×B =

{max- |A×B , if A 6= B,

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

-

6

π1 orπ2

π1 orπ2

max-

max-


-

6

1 2 3 4

1

2

3

4

< < <

s s s ss s s ss s s ss s s s��

1

2

3

4

-

6

2 4 3 1

2

4

3

1

∼ ≺ ≺

s s s ss s s ss s s ss s s s2

4

3

1

Order-preservable operations

Definition.

F : X 2 → X is said to be ≤-preserving for some total ordering ≤ on X iffor any x , y , x ′, y ′ ∈ X such that x ≤ x ′ and y ≤ y ′, we haveF (x , y) ≤ F (x ′, y ′)

Definition.

We say that F : X 2 → X is order-preservable if it is ≤-preserving forsome ≤

Q: Given an associative and quasitrivial F , is it order-preservable?


Definition.


Definition.




Definition.


Definition.




(∗)

-

6

π1 orπ2

π1 orπ2

max-

max-

2-quasilinearity : a ≺ b ∼ c ∼ d =⇒ a, b, c, d are not pairwise distinct

Theorem


F is associative, quasitrivial, and order-preservable

m

∃ - : F is of the form (∗) and - is 2-quasilinear


(∗)

-

6

π1 orπ2

π1 orπ2

max-

max-


Theorem



m



(∗)

-

6

π1 orπ2

π1 orπ2

max-

max-


Theorem



m



-

6

1 2 3 4

1

2

3

4

< < <

r r r rr r r rr r r rr r r r��

1

2

3

4

-

6

2 4 3 1

2

4

3

1

∼ ≺ ≺

r r r rr r r rr r r rr r r r2

4

3

1

-

6

1 2 3 4

1

2

3

4

< < <

r r r rr r r rr r r rr r r r� �

1

2

3

4

-

6

2 3 4 1

2

3

4

1

≺ ∼ ∼

r r r rr r r rr r r rr r r r� �� 2

3

4

1


-

6

1 2 3 4

1

2

3

4

< < <

r r r rr r r rr r r rr r r r��

1

2

3

4

-

6

2 4 3 1

2

4

3

1

∼ ≺ ≺

r r r rr r r rr r r rr r r r2

4

3

1

-

6

1 2 3 4

1

2

3

4

< < <

r r r rr r r rr r r rr r r r� �

1

2

3

4

-

6

2 3 4 1

2

3

4

1

≺ ∼ ∼

r r r rr r r rr r r rr r r r� �� 2

3

4

1

Final remarks

In arXiv: 1811.11113 and Quasitrivial semigroups: characterizations andenumerations (Semigroup Forum, 2018)

1 Characterizations and classifications of quasitrival semigroups by meansof certain equivalence relations

2 Characterization of associative, quasitrivial, and order-preservingoperations by means of single-plateauedness

3 New integer sequences (http://www.oeis.org)

Number of quasitrivial semigroups: A292932Number of associative, quasitrivial, and order-preservingoperations: A293005Number of associative, quasitrivial, and order-preservableoperations: Axxxxxx. . .

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

M. Couceiro, J. Devillet, and J.-L. Marichal.Quasitrivial semigroups: characterizations and enumerations.Semigroup Forum, In press. arXiv:1709.09162.

J. Devillet, J.-L. Marichal, and B. Teheux.Classifications of quasitrivial semigroups.arXiv:1811.11113.

H. Langer.The free algebra in the variety generated by quasi-trivial semigroups.Semigroup Forum, 20:151–156, 1980.

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Characterizations and classifications of quasitrivial ...€¦ · Characterizations and classi cations of quasitrivial semigroups Jimmy Devillet University of Luxembourg Luxembourg