Characterizations of nondecreasing semilattice
operations on chains
AAA96
Jimmy Devillet
Motivation
Let X be a nonempty set
Definition F : X2 → X is said to be idempotent if F (x , x ) = x x ∈ X quasitrivialif F (x , y ) ∈ {x , y } x , y ∈ X
≤-preserving for some total order ≤ on X if
Motivation
Fact. F is associative, quasitrivial, and commutative iff there exists a total order ≤ on X such that F = ∨.
Example. On X = {1, 2, 3, 4}, consider ≤ and ≤0
Motivation
≤ • 1 • 2 • 3 • 4 ≤0 • 3 • 1 • 2 • 4∨0(1, 2) = 2 and ∨0(1, 3) = 1 ⇒ ∨0 is not ≤-preserving
Single-peakedness
Definition. (Black, 1948) ≤0 is said to be single-peaked for ≤if for all a, b, c ∈ X , a ≤ b ≤ c =⇒ b ≤0 a ∨0c ∈ {a, c} ≤ • 1 • 2 • 3 • 4 ≤0 • 3 • 1 • 2 • 4
Single-peakedness
Definition. (Black, 1948) ≤0 is said to be single-peaked for ≤if for all a, b, c ∈ X , a ≤ b ≤ c =⇒ b ≤0 a ∨0c ∈ {a, c} ≤ • 1 • 2 • 3 • 4 ≤0 • 2 • 1 • 3 • 4
Single-peakedness
Definition. (Black, 1948) ≤0 is said to be single-peaked for ≤if for all a, b, c ∈ X ,
a ≤ b ≤ c =⇒ b ≤0 a ∨0c ∈ {a, c}
F is associative, quasitrivial, and commutative iff F = ∨
Theorem (Devillet et al., 2017)
For any F : X2 → X , the following are equivalent.
(i) F is associative, quasitrivial, commutative, and ≤-preserving
(ii) F = ∨0 for some ≤0 that is single-peaked for ≤
Towards a generalization
≤ will denote a total order on X will denote a join-semilattice order on X
F is associative, idempotent, and commutative iff there exists such that F = g.
Example. On X = {1, 2, 3, 4}, consider ≤ and
Towards a generalization
≤ • 1 • 2 • 3 • 4 • 1 • 4 • 3 • 2 • 4CI-property
Definition. We say that has theconvex-ideal property
(CI-propertyfor short) for ≤if for all a, b, c ∈ X ,
a ≤ b ≤ c =⇒ b a g c
Proposition
The following are equivalent.
(i) has the CI-property for ≤
CI-property
Definition. We say that has theCI-property for ≤if for all a, b, c ∈ X , a ≤ b ≤ c =⇒ b a g c ≤ • 1 • 2 • 3 • 4 • 1 • 4 • 3 • 2 • 4
CI-property
Definition. We say that has theCI-property for ≤if for all a, b, c ∈ X , a ≤ b ≤ c =⇒ b a g c ≤ • 1 • 2 • 3 • 4 • 1 • 3 • 4 • 2 • 3
CI-property
≤ • 1 • 2 • 3 • 4 • 1 • 3 • 4 •2 • 3Internality
Definition. F : X2 → X is said to beinternal if x ≤ F (x , y ) ≤ y for every x , y ∈ X with x ≤ y
Definition. We say that isinternal for ≤ if for all a, b, c ∈ X ,
a < b < c =⇒ (a 66= b g c and c 6= a g b)
Proposition
The following are equivalent.
(i) is internal for ≤
Internality
Definition. We say that isinternal for ≤ if for all a, b, c ∈ X ,
a < b < c =⇒ (a 66= b g c and c 6= a g b) ≤ • 1 • 2 • 3 • 4 • 1 • 3 • 4 •2 • 3
Internality
Definition. We say that isinternal for ≤ if for all a, b, c ∈ X , a < b < c =⇒ (a 66= b g c and c 6= a g b) ≤ • 1 • 2 • 3 • 4 • 1 • 2 • 4 •3 • 2
has the CI-property and is internal for ≤
Nondecreasingness
Definition. We say that isnondecreasing for ≤if CI-property for ≤
internal for ≤.
F is associative, idempotent, and commutative iff F = g
Theorem
For any F : X2 → X , the following are equivalent.
(i) F is associative, idempotent, commutative, and ≤-preserving
Finite case
Assume that X = {1, . . . , n}, is endowed with the usual total order
1 < . . . < n
Proposition
Finite case
By abinary treewe mean an unordered rooted tree in which every vertex has at most two children.
Proposition
The following are equivalent.
(i) is nondecreasing for ≤
Finite case
PropositionThe following are equivalent.
(i) is nondecreasing for ≤
(ii) The Hasse diagram of (X , ) is a binary tree satisfying (∗)
Selected references
D. Black.
On the rationale of group decision-making. J Polit Economy, 56(1):23–34, 1948 D. Black.
The theory of committees and elections. Kluwer Academic Publishers, Dordrecht, 1987. J. Devillet, G. Kiss and J.-M. Marichal.
Characterizations of quasitrivial symmetric nondecreasing associative operations. arXiv:1705.00719.
J. Devillet and B. Teheux.
Associative, idempotent, symmetric, and nondecreasing operations on chains. arXiv:1805.11936.
N. Kimura.
The structure of idempotent semigroups. I. Pacific J. Math., 8:257–275, 1958. D. McLean.
Idempotent semigroups.
