On associative, idempotent, symmetric, and nondecreasing operations
ISAS 2018
Jimmy Devillet
in collaboration with Bruno Teheux University of Luxembourg
Motivation
For any integern≥1, let Xn={1, ...,n} endowed with ≤ Definition
F:Xn2 →Xn is idempotent if
F(x,x) = x nondecreasing for ≤if
F(x,y) ≤ F(x0,y0) whenever x ≤ x0 and y ≤ y0
Motivation
Definition
e∈Xn is a neutral element of F if
F(x,e) = F(e,x) = x x ∈Xn
Motivation
Definition
F:Xn2 →Xn is a discrete uninorm if it is associative, symmetric, nondecreasing, and has a neutral element.
Idempotent discrete uninorms have been investigated for a few decades by several authors.
In the previous talk, the following characterization was provided.
Motivation
Definition. (Black, 1948)
≤0 is said to be single-peaked for≤if for alla,b,c ∈Xn, a≤b ≤c =⇒ b ≤0 a or b ≤0 c
Theorem(Couceiro et al., 2018)
For anyF:Xn2 →Xn, the following are equivalent.
(i) F is an idempotent discrete uninorm
(ii) F = max≤0 for some≤0 that is single-peaked for ≤
How can we generalize this result by removing the existence of a neutral element?
Towards a generalization
will denote a join-semilattice order on Xn
F is associative, idempotent, and symmetric iff there exists such thatF =g.
Example. OnX4={1,2,3,4}, consider ≤and
≤ 1• 2• 3• 4•
1•
4• 3•
•2 4•
Towards a generalization
≤ 1• 2• 3• 4•
1•
4• 3•
•2 4•
g(1,4) = 4 and g(3,4) = 3 ⇒ g is not nondecreasing What are thefor which g are nondecreasing?
CI-property
Definition. We say that has theconvex-ideal property (CI-propertyfor short) for ≤if for alla,b,c ∈Xn,
a≤b ≤c =⇒ bagc
Proposition
The following are equivalent.
(i) has the CI-property for ≤
(ii) Every ideal of (Xn,) is a convex subset of (Xn,≤)
CI-property
Definition. We say that has theCI-property for≤if for all a,b,c ∈Xn,
a≤b ≤c =⇒ bagc
≤ 1• 2• 3• 4•
1•
4• 3•
•2 4•
does not have the CI-property for≤
CI-property
Definition. We say that has theCI-property for≤if for all a,b,c ∈Xn,
a≤b ≤c =⇒ bagc
≤ 1• 2• 3• 4•
1•
3• 4•
•2 3•
has the CI-property for ≤
CI-property
≤ 1• 2• 3• 4•
1•
3• 4•
•2 3•
g(1,2) = 3 and g(2,2) = 2 =⇒ g is not nondecreasing
Internality
Definition. F:Xn2 →Xnis said to be internal ifx ≤F(x,y)≤y for everyx,y ∈Xnwith x ≤y
Definition. We say that isinternal for ≤if for alla,b,c ∈Xn, a<b<c =⇒ (a66=bgc and c 6=agb) Proposition
The following are equivalent.
(i) is internal for ≤
(ii) The join operation g ofis internal
A direct consequence
Corollary
For anyF:Xn2 →Xn, the following are equivalent.
(i) F is associative, internal, and symmetric (ii) F =gfor some that is internal for≤
Fact. If F is idempotent and nondecreasing then it is internal.
Internality
Definition. We say that isinternal for ≤if for alla,b,c ∈Xn, a<b<c =⇒ (a66=bgc and c 6=agb)
≤ 1• 2• 3• 4•
1•
3• 4•
•2 3•
has the CI-property but is not internal for≤
Internality
Definition. We say that isinternal for ≤if for alla,b,c ∈Xn, a<b<c =⇒ (a66=bgc and c 6=agb)
≤ 1• 2• 3• 4•
1•
2• 4•
•3 2•
has the CI-property and is internal for≤ Also,g is nondecreasing
Nondecreasingness
Definition. We say that isnondecreasing for ≤if CI-property for≤
internal for ≤.
F is associative, idempotent, and symmetric iffF =g
Theorem
For anyF:Xn2 →Xn, the following are equivalent.
(i) F is associative, idempotent, symmetric, and nondecreasing (ii) F =gfor some that is nondecreasing for≤
Enumeration
Proposition
The number of associative, idempotent, symmetric, and nondecreas- ing operations onXnis the nth Catalan number.
Final remarks
1 Our characterization is still valid on arbitrary chains (not necessarily finite)
2 In arXiv: 1805.11936, we provided an alternative characterization of the class of associative, idempotent, symmetric, and nondecreasing operations onXn.
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.
M. Couceiro, J. Devillet and J.-L. Marichal.
Chracterizations of idempotent discrete uninorms, Fuzzy Sets and Syst., 334:60–72, 2018.
B. De Baets, J. Fodor, D. Ruiz-Aguilera, and J. Torrens.
Idempotent uninorms on finite ordinal scales.
Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 17(1):1–14, 2009.
J. Devillet and B. Teheux.
Associative, idempotent, symmetric, and nondecreasing operations on chains.
arXiv:1805.11936.
G. Gr¨atzer.
General Lattice Theory, Second edition.
Birkh¨auser, 2003
