On quasitrivial semigroups

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On quasitrivial semigroups

Jimmy Devillet

University of Luxembourg

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Part I: Single-plateauedness

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Weak orderings

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

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

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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

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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

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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 besingle-plateaued for ≤if

a_i< a_j < a_k =⇒ a_j ≺ a_i or a_j ≺ a_k or a_i ∼ a_j ∼ a_k

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 A

A

@@ r

r r r

r r

- 6

a₁ a₂ a₃ a₄ a₅ a₆ a₆

a₅ a1∼ a2

a3∼ a4

A A

A

@@ r r

r r

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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 besingle-plateaued for ≤if

a_i< a_j < a_k =⇒ a_j ≺ a_i or a_j ≺ a_k or a_i ∼ a_j ∼ a_k

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 A

A

@@ r

r r r

r r

- 6

a₁ a₂ a₃ a₄ a₅ a₆ a₆

a₅ a1∼ a2

a3∼ a4

A A

A

@@ r r

r r

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Part II: Quasitrivial semigroups

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Quasitriviality

Definition

F : X²→ X is said to bequasitrivial (orconservative) if F (x , y ) ∈ {x , y } x , y ∈ X

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

r r r

- 6

1 2 3

1 2

3

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Quasitriviality

Definition

r r r

- 6

1 2 3

1 2

3

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Quasitriviality

Definition

r r r

- 6

1 2 3

1 2

3

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Projections

Definition.

Theprojection operationsπ1: X²→ X and π2: X²→ X are respectively defined by

π1(x , y ) = x , x , y ∈ X π₂(x , y ) = y , x , y ∈ X

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Quasitrivial semigroups

Theorem (L¨anger, 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 / ∼

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π₁or π2

π₁or π₂ max_-

max_-

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Quasitrivial semigroups

Theorem (L¨anger, 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

π₁or π2

π₁or π₂ max_-

max_-

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Quasitrivial semigroups

- 6

1 2 3 4

< < <

s s s s

1

2 3

4

- 6

2 4 3 1

∼ ≺ ≺

s s s s

2 4

3 1

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Order-preserving operations

Definition.

F : X²→ X is said to be≤-preservingfor some total ordering ≤ on X if for any x , y , x⁰, y⁰∈ X such that x ≤ x⁰ and y ≤ y⁰, we have

F (x , y ) ≤ F (x⁰, y⁰)

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Order-preserving operations

(∗)

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π1 or π2

π1or π2

max_-

single-plateauedness: a < b < c =⇒ a ≺ b or a ≺ c or a ∼ b ∼ c

Theorem (Couceiro et al., 2018)

F is associative, quasitrivial, and ≤-preserving m

∃ - : F is of the form (∗) and - is single-plateaued for ≤

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Order-preserving operations

(∗)

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π1 or π2

π1or π2

max_-

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Order-preserving operations

(∗)

- 6

π1 or π2

π1or π2

max_-

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Order-preserving operations

- 6

1 2 3 4

< < <

r r r r

1

2 3

4

- 6

2 4 3 1

∼ ≺ ≺

r r r r

2 4

3 1

- 6

1 2 3 4

< < <

r r r r

1 2

3 4

- 6

2 3 4 1

≺ ∼ ∼

r r r r

2 3

4 1

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Order-preserving operations

- 6

1 2 3 4

< < <

r r r r

1

2 3

4

- 6

2 4 3 1

∼ ≺ ≺

r r r r

2 4

3 1

- 6

1 2 3 4

< < <

r r r r

1 2

3 4

- 6

2 3 4 1

≺ ∼ ∼

r r r r

2 3

4 1

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Quasitrivial n-ary semigroups

Definition

F : Xⁿ→ X is said to be quasitrivialif

F (x1, . . . , xn) ∈ {x1, . . . , xn} x1, . . . , xn∈ X

associativeif

F (x1, . . . , xi −1, F (xi, . . . , xi +n−1), xi +n, . . . , x2n−1)

= F (x1, . . . , xi, F (xi +1, . . . , xi +n), xi +n+1, . . . , x2n−1) for all x₁, . . . , x_2n−1∈ X and all 1 ≤ i ≤ n − 1.

Characterization?

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Quasitrivial n-ary semigroups

Definition

F (x1, . . . , xn) ∈ {x1, . . . , xn} x1, . . . , xn∈ X

associativeif

Characterization?

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Quasitrivial n-ary semigroups

Definition

F (x1, . . . , xn) ∈ {x1, . . . , xn} x1, . . . , xn∈ X

associativeif

Characterization?

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Quasitrivial n-ary semigroups

Definition

F (x1, . . . , xn) ∈ {x1, . . . , xn} x1, . . . , xn∈ X

associativeif

Characterization?

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Reducibility

Definition

F : Xⁿ→ X and G : X² → X associative operations.

F is said to bereducible to G if

F (x₁, . . . , x_n) = G (x₁, G (x₂, G (. . . , G (x_n−1, x_n) . . .)))

Example. On X = R

F (x1, x2, x3, x4, x5) = x1+ x2+ x3+ x4+ x5, and

G (x , y ) = x + y

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Reducibility

Definition

F (x₁, . . . , x_n) = G (x₁, G (x₂, G (. . . , G (x_n−1, x_n) . . .)))

Example. On X = R

F (x1, x2, x3, x4, x5) = x1+ x2+ x3+ x4+ x5, and

G (x , y ) = x + y

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Reducibility

Definition

F (x₁, . . . , x_n) = G (x₁, G (x₂, G (. . . , G (x_n−1, x_n) . . .)))

Example. On X = R

F (x1, x2, x3, x4, x5) = x1+ x2+ x3+ x4+ x5, and

G (x , y ) = x + y

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Neutral elements

Definition

e ∈ X is said to be aneutral element for F if

F (x , e, . . . , e) = F (e, x , e, . . . , e) = . . . = F (e, . . . , e, x ) = x , for all x ∈ X

Example. F (x₁, . . . , x_n) =Pn

i =1x_i (mod n − 1) Proposition (Couceiro and D., 2019)

Any quasitrivial n-ary semigroup has at most two neutral elements.

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Neutral elements

Definition

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Neutral elements

Definition

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Quasitrivial n-ary semigroups

Theorem (Couceiro and D., 2019)

Any quasitrivial n-ary semigroup is reducible to asemigroup.

But the binary reduction is not necessarily quasitrivial nor unique.

Example.

F (x , y , z) = x + y + z (mod 2)

G (x , y ) = x + y (mod 2) G⁰(x , y ) = x + y + 1 (mod 2)

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Quasitrivial n-ary semigroups

Example.

F (x , y , z) = x + y + z (mod 2)

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Quasitrivial n-ary semigroups

Example.

F (x , y , z) = x + y + z (mod 2)

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Uniqueness of a reduction

Proposition (Couceiro and D., 2019)

F : Xⁿ→ X associative and quasitrivial. TFAE (i) F has a unique binary reduction

(ii) F has a quasitrivial binary reduction (iii) F has an idempotent binary reduction (iv) F has at most one neutral element

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Uniqueness of a reduction

Proposition (Couceiro and D., 2019)

F : Xⁿ→ X associative and quasitrivial. TFAE (i) F has a unique binary reduction

(ii) F has a quasitrivial binary reduction (iii) F has an idempotent binary reduction (iv) F has at most one neutral element

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On quasitrivial semigroups