Characterizations and classifications of quasitrivial semigroups
Jimmy Devillet
University of Luxembourg Luxembourg
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 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 Weak orderings
Recall that a weak ordering (or total 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 Weak orderings
Recall that a weak ordering (or total 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 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 6 a3 ∼ a4 a3 ∼ a4 A A a2 r r a1 ∼ a2 r r A A a1 ∼ a5 r A a5 r r A @ @ a6 r r @ a6 r @ r - r - a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 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 6 a3 ∼ a4 a3 ∼ a4 A A a2 r r a1 ∼ a2 r r A A a1 ∼ a5 r A a5 r r A @ @ a6 r r @ a6 r @ r - r - a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 Single-plateaued weak orderings
Q: Given - is it possible to find ≤ for which - is single-plateaued?
0 Example: On X = {a1, a2, a3, a4} consider - and - defined by
0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4
Yes! Consider ≤ defined by a3 < a1 < a2 < a4
6
a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 No! Single-plateaued weak orderings
Q: Given - is it possible to find ≤ for which - is single-plateaued?
0 Example: On X = {a1, a2, a3, a4} consider - and - defined by
0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4
Yes! Consider ≤ defined by a3 < a1 < a2 < a4
6
a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 No! Single-plateaued weak orderings
Q: Given - is it possible to find ≤ for which - is single-plateaued?
0 Example: On X = {a1, a2, a3, a4} consider - and - defined by
0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4
Yes! Consider ≤ defined by a3 < a1 < a2 < a4
6
a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 No! Single-plateaued weak orderings
Q: Given - is it possible to find ≤ for which - is single-plateaued?
0 Example: On X = {a1, a2, a3, a4} consider - and - defined by
0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4
Yes! Consider ≤ defined by a3 < a1 < a2 < a4
6
a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 No! Single-plateaued weak orderings
Q: Given - is it possible to find ≤ for which - is single-plateaued?
0 Example: On X = {a1, a2, a3, a4} consider - and - defined by
0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4
Yes! Consider ≤ defined by a3 < a1 < a2 < a4
6
a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 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 Assume the axiom of choice.
- 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 ≤
6 3 3 r r r 2 2 r r r 1 1 r r r - 1 2 3 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 ≤
6 3 3 r r r 2 2 r r r 1 1 r r r - 1 2 3 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 ≤
6 3 3 r r r 2 2 r r r 1 1 r r r - 1 2 3 Projections
Definition. 2 2 The projection operations π1 : X → X and π2 : X → X are respectively defined by
π1(x, y) = x, x, y ∈ X
π2(x, y) = y, x, y ∈ X Quasitrivial semigroups
Theorem (L¨anger,1980)
F is associative and quasitrivial m ( max | , if A 6= B, - A×B ∃ - : F |A×B = ∀A, B ∈ X / ∼ π1|A×B or π2|A×B , if A = B,
6
max - π1 or π2
π1 or max - π2 - Quasitrivial semigroups
Theorem (L¨anger,1980)
F is associative and quasitrivial m ( max | , if A 6= B, - A×B ∃ - : F |A×B = ∀A, B ∈ X / ∼ π1|A×B or π2|A×B , if A = B,
6
max - π1 or π2
π1 or max - π2 - Quasitrivial semigroups
6 6 4 1 4 1 3 3 3 s§ s s s¤ 3 s s s s 2 4 2 s s s s 4 s s s s 1 2 1 s¦ s s s¥ 2 s s s s s s s s - s s s s - 1< 2 < 3 < 4 2∼ 4≺ 3 ≺ 1 Order-preservable operations
Definition. F : X 2 → X is said to be ≤-preserving for some total ordering ≤ on X if for any x, y, x 0, y 0 ∈ X such that x ≤ x 0 and y ≤ y 0, we have F (x, y) ≤ F (x 0, y 0)
Definition. We say that F : X 2 → X is order-preservable if it is ≤-preserving for some ≤
Q: Given an associative and quasitrivial F , is it order-preservable? Order-preservable operations
Definition. F : X 2 → X is said to be ≤-preserving for some total ordering ≤ on X if for any x, y, x 0, y 0 ∈ X such that x ≤ x 0 and y ≤ y 0, we have F (x, y) ≤ F (x 0, y 0)
Definition. We say that F : X 2 → X is order-preservable if it is ≤-preserving for some ≤
Q: Given an associative and quasitrivial F , is it order-preservable? Order-preservable operations
Definition. F : X 2 → X is said to be ≤-preserving for some total ordering ≤ on X if for any x, y, x 0, y 0 ∈ X such that x ≤ x 0 and y ≤ y 0, we have F (x, y) ≤ F (x 0, y 0)
Definition. We say that F : X 2 → X is order-preservable if it is ≤-preserving for some ≤
Q: Given an associative and quasitrivial F , is it order-preservable? Order-preservable operations
6
max - π1 or π (∗) 2
π1 or max π2 - - 2-quasilinearity : a ≺ b ∼ c ∼ d =⇒ a, b, c, d are not pairwise distinct
Theorem Assume the axiom of choice.
F is associative, quasitrivial, and order-preservable
m
∃ - : F is of the form (∗) and - is 2-quasilinear Order-preservable operations
6
max - π1 or π (∗) 2
π1 or max π2 - - 2-quasilinearity : a ≺ b ∼ c ∼ d =⇒ a, b, c, d are not pairwise distinct
Theorem Assume the axiom of choice.
F is associative, quasitrivial, and order-preservable
m
∃ - : F is of the form (∗) and - is 2-quasilinear Order-preservable operations
6
max - π1 or π (∗) 2
π1 or max π2 - - 2-quasilinearity : a ≺ b ∼ c ∼ d =⇒ a, b, c, d are not pairwise distinct
Theorem Assume the axiom of choice.
F is associative, quasitrivial, and order-preservable
m
∃ - : F is of the form (∗) and - is 2-quasilinear Order-preservable operations
6 4 6 1 4 1 3 3 3 r§ r r r¤ 3 r r r r 2 4 2 r r r r 4 r r r r 1 2 1 r¦ r r r¥ 2 r r r r r r r r - r r r r - 1< 2 < 3 < 4 2∼ 4≺ 3 ≺ 1
6 4 6 1 4 1 § 3 ¤ § 4 ¤ 3 r r r r 4 r r r r 2 § 3 ¤ 2 r r r r 3 r r r r 1 2 1 r r r r 2 r r r r r r r r - r r r r - 1< 2 < 3 < 4 2≺ 3∼ 4 ∼ 1 Order-preservable operations
6 4 6 1 4 1 3 3 3 r§ r r r¤ 3 r r r r 2 4 2 r r r r 4 r r r r 1 2 1 r¦ r r r¥ 2 r r r r r r r r - r r r r - 1< 2 < 3 < 4 2∼ 4≺ 3 ≺ 1
6 4 6 1 4 1 § 3 ¤ § 4 ¤ 3 r r r r 4 r r r r 2 § 3 ¤ 2 r r r r 3 r r r r 1 2 1 r r r r 2 r r r r r r r r - r r r r - 1< 2 < 3 < 4 2≺ 3∼ 4 ∼ 1 Final remarks
In arXiv: 1811.11113 and Quasitrivial semigroups: characterizations and enumerations (Semigroup Forum, 2018)
1 Characterizations and classifications of quasitrival semigroups by means of certain equivalence relations
2 Characterization of associative, quasitrivial, and order-preserving operations by means of single-plateauedness
3 New integer sequences (http://www.oeis.org) Number of quasitrivial semigroups: A292932 Number of associative, quasitrivial, and order-preserving operations: A293005 Number of associative, quasitrivial, and order-preservable operations: 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. L¨anger. The free algebra in the variety generated by quasi-trivial semigroups. Semigroup Forum, 20:151–156, 1980.