On bisymmetric and quasitrivial operations 56th ISFE Jimmy Devillet

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On bisymmetric and quasitrivial operations

56th ISFE

Jimmy Devillet

University of Luxembourg

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Connectedness and Contour Plots

LetX be a nonempty set and letF: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 notF-connected to another point in X²

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Connectedness and Contour Plots

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

r r r r

- 6

1 2 3 4

1 2

3 4

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Bisymmetry and quasitriviality

Definition

F:X² →X is said to be bisymmetric if

F(F(x,y),F(u,v)) = F(F(x,u),F(y,v)) x,y,u,v ∈X quasitrivialif

F(x,y) ∈ {x,y} x,y ∈X

Lemma(Kepka, 1981)

IfF is bisymmetric and quasitrivial then it is associative

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Weak orderings

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

transitiveif ∀x,y,z: xRy andyRz impliesxRz

Aweak ordering on X is a binary relation -onX that is total and transitive.

symmetric part: ∼ asymmetric part: ≺

Recall that∼is an equivalence relation on X and that≺induces a linear ordering on the quotient setX/∼

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Motivation

Theorem(L¨anger, 1980)

F is associative and quasitrivial iff there exists a weak ordering - onX such that

F|_A×B =

(max_-|_A×B, if A6=B,

π₁|_A×B or π₂|_A×B, if A=B, ∀A,B ∈X/∼ IfX =Xn={1, . . . ,n}, then

x - y ⇐⇒ |F⁻¹({x})| ≤ |F⁻¹({y})|

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Motivation

F|_A×B =

π1|_A×B or π2|_A×B, if A=B, ∀A,B ∈X/∼

x - y ⇐⇒ |F⁻¹({x})| ≤ |F⁻¹({y})|

r r r

1 2

3

1 < 2 < 3

r r r

2 1

3

2 ≺ 1 ∼ 3

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Motivation

s s s

1 2

3

1 < 2 < 3

s s s

2 1

3

2 ≺ 1 ∼ 3

F(F(2,3),F(1,2)) = 3 6= 1 = F(F(2,1),F(3,2))

=⇒ F is not bisymmetric

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Quasilinearity

Let -be a weak ordering onX

Definition

-isquasilinear if there exist no pairwise distinct a,b,c ∈X such thata≺b∼c

Example. OnX ={1,2,3,4}, consider the- 2∼3≺1≺4

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Quasilinearity

F|_A×B =

s s s

1 2

3

1 < 2 < 3

s s s

2 1

3

2 ≺ 1 ∼ 3

-is not quasilinear and F is not bisymmetric

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Quasilinearity

x - y ⇐⇒ |F⁻¹({x})| ≤ |F⁻¹({y})|

s s s

1 2

3

1 < 2 < 3

s s s

1 3

2

1 ∼ 3 ≺ 2

-is quasilinear and F is bisymmetric

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A characterization

F|_A×B =

(max_-|_A×B, ifA6=B,

π₁|_A×B orπ₂|_A×B, ifA=B, ∀A,B ∈X/∼ (∗)

Theorem

For anyF:X² →X, the following are equivalent.

(i) F is bisymmetric and quasitrivial

(ii) F is of the form (∗) for some quasilinear -

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A characterization

We denote theset of minimal elements of X for -by min_-X

π1 orπ2

max_-

min_-X X\min_-X min_-X

X \min_-X

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Bisymmetric and quasitrivial operations

r r r r r r r r r

r r r

r r r r r r

r r r r r r r r r

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The nondecreasing case

Let≤ be a linear ordering onX

Definition. F:X² →X is nondecreasing for ≤if

F(x,y)≤F(x⁰,y⁰) whenever x≤x⁰ and y≤y⁰

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The nondecreasing case

F|_A×B =

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(Couceiro et al., 2018)

-is said to beweakly single-peaked for ≤if for any a,b,c ∈X, a<b<c =⇒ 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 for≤

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Weakly single-peaked weak orderings

x - y ⇐⇒ |F⁻¹({x})| ≤ |F⁻¹({y})|

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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Associative quasitrivial and nondecreasing operations

F|_A×B =

Theorem(Couceiro et al., 2018)

(i) F is associative, quasitrivial, and nondecreasing

(ii) F is of the form (∗) for some -that is weakly single-peaked for ≤

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Bisymmetric quasitrivial and nondecreasing operations

F|_A×B =

Theorem

(i) F is bisymmetric, quasitrivial, and nondecreasing

(ii) F is of the form (∗) for some -that is quasilinear and weakly single-peaked for≤

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Final remarks

In arXiv: 1712.07856

1 Characterizations of the class of bisymmetric and quasitrivial operations as well as the subclass of those operations that are nondecreasing

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

Number of bisymmetric and quasitrivial operations: A296943 Number of bisymmetric, quasitrivial, and nondecreasing operations: A296953

. . .

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Selected references

N. L. Ackerman.

A characterization of quasitrivialn-semigroups.

To appear inAlgebra Universalis.

M. Couceiro, J. Devillet, and J.-L. Marichal.

Quasitrivial semigroups: characterizations and enumerations.

Semigroup forum, In press. arXiv:1709.09162.

J. Devillet.

Bisymmetric and quasitrivial operations: characterizations and enumerations.

arXiv: 1712.07856.

Z. Fitzsimmons.

Single-peaked consistency for weak orders is easy.

In Proc. of the 15th Conf. on Theoretical Aspects of Rationality and Knowledge (TARK 2015), pages 127–140, June 2015. arXiv:1406.4829.

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.