Commuting Polynomial Operations of Distributive Lattices

Behrisch, Mike; Couceiro, Miguel; Kearnes, Keith A.; Lehtonen, Erkko; Szendrei, Ágnes

doi:10.1007/s11083-011-9231-3

Cited by 6 publications

(13 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus each polynomial function p has a unique DNF with monotone coefficients. We call such a DNF a maximal disjunctive normal form, as the coefficients of this DNF are the greatest among all DNF's representing the same polynomial function p (see [1]). In the sequel we will always consider lattice polynomial functions in maximal DNF.…”

Section: Preliminariesmentioning

confidence: 99%

“…(We omit the analogous discussion of the cases where L has one boundary element.) Polynomial functions over L can still be given in DNF of the form (2.1) by allowing the coefficients c I to take also the values 0 and 1, which are considered as external boundary elements (see, e.g., [1,3]). For example, a polynomial function p(x, y) = a ∨ x ∨ (b ∧ x ∧ y) can be rewritten as…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Interpolation by polynomial functions of distributive lattices: a generalization of a theorem of R. L. Goodstein

Couceiro

Waldhauser

2013

Algebra Univers.

Self Cite

View full text Add to dashboard Cite

We consider the problem of interpolating functions partially defined over a distributive lattice, by means of lattice polynomial functions. Goodstein's theorem solves a particular instance of this interpolation problem on a distributive lattice L with least and greatest elements 0 and 1, resp.: Given a function f : {0, 1} n → L, there exists a lattice polynomial function p : L n → L such that p| {0,1} n = f if and only if f is monotone; in this case, the interpolating polynomial p is unique. We extend Goodstein's theorem to a wider class of partial functions f : D → L over a distributive lattice L, not necessarily bounded, and where D ⊆ L n is allowed to range over n-dimensional rectangular boxes D = {a 1 , b 1 } × • • • × {an, bn} with a i , b i ∈ L and a i < b i , and determine the class of such partial functions which can be interpolated by lattice polynomial functions. In this wider setting, interpolating polynomials are not necessarily unique; we provide explicit descriptions of all possible lattice polynomial functions which interpolate these partial functions, when such an interpolation is available.

show abstract

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Interpolation by polynomial functions of distributive lattices: a generalization of a theorem of R. L. Goodstein

Couceiro

Waldhauser

2013

Algebra Univers.

Self Cite

View full text Add to dashboard Cite

show abstract

“…For which capacities do the S-integrals commute for all functions u? The first question is considered by Narukawa and Torra [12] for more general fuzzy integrals, and the second one by Behrisch et al [1], albeit in the larger setting of distributive lattices, for general lattice polynomials. However, in our paper, we only consider a totally ordered set L. It is of interest to adapt these results for S-integrals valued on totally ordered sets, as they become more palatable.…”

Section: The Commutation Of Sugeno Integralsmentioning

confidence: 99%

“…Proof: The proof is inspired by a paper on the commutation of polynomials on distributive lattices [1], and requires several lemmas listed below. Our proof is easier to read and simpler, though.…”

Section: Otherwisementioning

confidence: 99%

See 1 more Smart Citation

Sugeno Integrals and the Commutation Problem

Dubois¹,

Fargier²,

Rico³

2018

Modeling Decisions for Artificial Intelligence

View full text Add to dashboard Cite

In decision problems involving two dimensions (like several agents and several criteria) the properties of expected utility ensure that the result of a multicriteria multiperson evaluation does not depend on the order with which the aggregations of local evaluations are performed (agents first, criteria next, or the converse). We say that the aggregations on each dimension commute. Ben Amor, Essghaier and Fargier have shown that this property holds when using pessimistic possibilistic integrals on each dimension, or optimistic ones, while it fails when using a pessimistic possibilistic integral on one dimension and an optimistic one on the other. This paper studies and completely solves this problem when Sugeno integrals are used in place of possibilistic integrals, indicating that there are capacities other than possibility and necessity measures that ensure commutation of Sugeno integrals.

show abstract