Characterizations of nondecreasing semilattice operations on chains

Transcription

1 Characterizations of nondecreasing semilattice operations on chains AAA96 Jimmy Devillet in collaboration with Bruno Teheux University of Luxembourg

2 Motivation Let X be a nonempty set Definition F : X 2 X is said to be idempotent if F (x, x) = x quasitrivial if F (x, y) {x, y} x X x, y X -preserving for some total order on X if F (x, y) F (x, y ) whenever x x and y y

3 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

4 Motivation (1, 2) = 2 and (1, 3) = 1 is not -preserving What are the for which are -preserving?

5 Single-peakedness Definition. (Black, 1948) is said to be single-peaked for if for all a, b, c X, a b c = b a c {a, c} is not single-peaked for

6 Single-peakedness Definition. (Black, 1948) is said to be single-peaked for if for all a, b, c X, a b c = b a c {a, c} is single-peaked for and is -preserving

7 Single-peakedness Definition. (Black, 1948) is said to be single-peaked for if for all a, b, c X, a b c = b a c {a, c} F is associative, quasitrivial, and commutative iff F = Theorem (Devillet et al., 2017) For any F : X 2 X, the following are equivalent. (i) F is associative, quasitrivial, commutative, and -preserving (ii) F = for some that is single-peaked for How can we generalize this result by relaxing quasitriviality into idempotency?

8 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 =. Example. On X = {1, 2, 3, 4}, consider and 2

9 Towards a generalization 2 (1, 4) = 4 and (3, 4) = 3 is not -preserving What are the for which are -preserving?

10 CI-property Definition. We say that has the convex-ideal property (CI-property for short) for if for all a, b, c X, a b c = b a c Proposition The following are equivalent. (i) has the CI-property for (ii) Every ideal of (X, ) is a convex subset of (X, )

11 CI-property Definition. We say that has the CI-property for if for all a, b, c X, a b c = b a c 2 does not have the CI-property for

12 CI-property Definition. We say that has the CI-property for if for all a, b, c X, a b c = b a c 2 has the CI-property for

13 CI-property 2 (1, 2) = 3 and (2, 2) = 2 = is not -preserving

14 Internality Definition. F : X 2 X is said to be internal if x F (x, y) y for every x, y X with x y Definition. We say that is internal for if for all a, b, c X, Proposition a < b < c = (a b c and c a b) The following are equivalent. (i) is internal for (ii) The join operation of is internal

15 Internality Definition. We say that is internal for if for all a, b, c X, a < b < c = (a b c and c a b) 2 has the CI-property but is not internal for

16 Internality Definition. We say that is internal for if for all a, b, c X, a < b < c = (a b c and c a b) 3 has the CI-property and is internal for Also, is -preserving

17 Nondecreasingness Definition. We say that is nondecreasing for if CI-property for internal for. F is associative, idempotent, and commutative iff F = Theorem For any F : X 2 X, the following are equivalent. (i) F is associative, idempotent, commutative, and -preserving (ii) F = for some that is nondecreasing for

18 Finite case Assume that X = {1,..., n}, is endowed with the usual total order 1 <... < n Proposition The number of nondecreasing join-semilattice orders on X is the n th Catalan number.

19 Finite case By a binary tree we mean an unordered rooted tree in which every vertex has at most two children. Proposition The following are equivalent. (i) is nondecreasing for (ii) The Hasse diagram of (X, ) is a binary tree satisfying ( )

20 Finite case Proposition The following are equivalent. (i) is nondecreasing for (ii) The Hasse diagram of (X, ) is a binary tree satisfying ( ) ( ): (X, )... (X, ).

21 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, J. Devillet, G. Kiss and J.-M. Marichal. Characterizations of quasitrivial symmetric nondecreasing associative operations. arxiv: J. Devillet and B. Teheux. Associative, idempotent, symmetric, and nondecreasing operations on chains. arxiv: N. Kimura. The structure of idempotent semigroups. I. Pacific J. Math., 8: , D. McLean. Idempotent semigroups. Amer. Math. Monthly, 61: , 1954.