Linear extensions and order-preserving poset partitions

We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving partitions) of a finite $n$-element poset $P$ with $n\geq 3$ is homotopy equivalent to a wedge of spheres of dimension $n-3$. If $P$ is connected, then the number of spheres is equal to the number of linear extensions of $P$. In general, the number of spheres is equal to the number of cyclic extensions of $P$

Distorted Copulas: Constructions and Tail Dependence

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1] the distortion C ψ: [0, 1]2 → [0, 1], C ψ(x, y) = ψ{C[ψ−1(x), ψ−1(y)]} is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ that ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails

Information boundedness principle in fuzzy inference process

summary:The information boundedness principle requires that the knowledge obtained as a result of an inference process should not have more information than that contained in the consequent of the rule. From this point of view relevancy transformation operators as a generalization of implications are investigated

Shuffles of copulas

none2We show that every copula that is a shuffle of Min can be described in terms of a push--forward of the doubly stochastic measure induced by the copula $M$ by means of a specific transformation of the unit square. This fact allows to generalize the notion of shuffle by replacing the measure induced by $M$ with an arbitrary doubly stochastic measure, and, hence, the copula $M$ by any copula $C$.Si tratta di una notevole estensione del concetto di shuffle basata su delicati risultati di teoria ergodicaF. DURANTE; P. SARKOCI; C. SEMPIDurante, Fabrizio; P., Sarkoci; Sempi, Carl

Remarks on Two Product-like Constructions for Copulas

summary:We investigate two constructions that, starting with two bivariate copulas, give rise to a new bivariate and trivariate copula, respectively. In particular, these constructions are generalizations of the $\ast$-product and the $\star$-product for copulas introduced by Darsow, Nguyen and Olsen in 1992. Some properties of these constructions are studied, especially their relationships with ordinal sums and shuffles of Min