Elsevier

Information Sciences

Volume 178, Issue 18, 15 September 2008, Pages 3557-3564
Information Sciences

Aggregation of infinite sequences

https://doi.org/10.1016/j.ins.2008.05.020Get rights and content

Abstract

Infinitary aggregation functions acting on sequences and possessing some a priori given properties as additivity, comonotone additivity, symmetry, etc., are investigated. On the other side, we discuss infinitary aggregation functions related to given extended aggregation functions, where special attention is given to triangular norms, triangular conorms and weighted means.

Introduction

The aggregation of a fixed finite number n of inputs from a given scale (chain of possible values) E is proceeded by means of some n-ary aggregation function A(n):EnE which is non-decreasing in each argument and for each tE there are x,yEn such thatA(n)(x)tA(n)(y).Observe that if E is a bounded chain with top element 1 and bottom element 0, then (1) is equivalent to the idempotency of 1 and 0, i.e., A(n)(1,,1)=1 and A(n)(0,,0)=0, compare [12], [14]. In the case of a finite but not a priori fixed number of inputs extended aggregation functions A:nNEnE are applied, where for each nN, A(n)=A|En is an n-ary aggregation function, and A(1):EE is the identity function, see e.g. [5], [15]. The aim of this paper is to discuss the aggregation of an infinite sequence of inputs, i.e., of non-decreasing functions A():ENE such that (1) (with x,yEN) is fulfilled. Obviously, such functions can be defined arbitrarily without any relation to the extended aggregation functions A:nNEnE. Throughout this paper, we will deal with the most common continuous scale E=[0,1]. Some other types of scales will be briefly mentioned in the concluding remarks.

The aggregation of infinitely but still countably many inputs is important in several mathematical areas, such as discrete probability theory, but also in non-mathematical areas, such as decision problems with an infinite jury, game theory with infinitely many players, etc. Though these theoretical tasks seem to be far from reality, they enable a better understanding of decision problems with an extremely huge jury, game theoretical problems with extremely many players, etc., see [9], [20], [22], [25].

We consider the set s=[0,1]N of all sequences x=(x1,x2,,xi,), where xi[0,1],iN. The input space s equipped with the standard Cartesian ordering, i.e., xy means xiyi,iN, is a complete lattice with bottom element 0={0}N and top element 1={1}N. We equip s with the coordinatewise convergence, i.e., a sequence x(n)=(x1(n),x2(n),,xi(n),) from s converges to x=(x1,x2,,xn,)s if and only if limnxi(n)=xi for all iN.

In Section 2, we will discuss infinitary aggregation functions on sequences possessing some a priori given properties, such as additivity, comonotone additivity, symmetry, etc. On the other side, in Section 3 we will discuss infinitary aggregation functions A():[0,1]N[0,1] related to a given extended aggregation function A:nN[0,1]n[0,1], where special attention is taken on triangular norms, triangular conorms and weighted means. Note that the discussion of the infinitary arithmetic mean AM():[0,1]N[0,1] can be found in [10], [11].

Section snippets

Infinitary aggregation functions and their properties

We extend the notion of the usual aggregation function to the case of infinite inputs.

Definition 1

A function A():s[0,1] is an (infinitary) aggregation function if it satisfies the following conditions:

  • (i)

    It is non-decreasing, i.e., xy implies A()(x)A()(y).

  • (ii)

    A()(0)=0 and A()(1)=1.

Properties of these functions are defined similarly as the corresponding properties of n-ary aggregation functions, see e.g. [5], [12], [14].

Definition 2

An aggregation function A():s[0,1] is said to be additive, comonotone additive,

Extended aggregation functions and infinitary aggregation functions

For a given extended aggregation function A:nN[0,1]n[0,1], we will look for an appropriate aggregation function A():s[0,1] somehow linked to A. A genuine approach is to define A() as a limit of (A(n))nNA()((xn)nN)=limnA(n)(x1,,xn).If this limit exists, for any (xn)nNs, we will accept A() given by (4) as an extension of A to the domain s and we will keep the notation A also for A() whenever appropriate. The aggregation function A will be called countably extendable.

A sufficient

Concluding remarks

Results of this paper are related to problems discussed, e.g., in [3]. Moreover, they can be applied to extend the results presented on finite domains, such as those in [18], [19], [24].

We have considered and discussed infinitary extensions of aggregation functions acting on s. Completely analogous considerations can be done on arbitrary closed (extended) real scale E=[a,b][-,]. In the case when E is not a closed real interval, the possible range of A() defined on EN should be extended to

Acknowledgements

Both authors want to thank for a support of the bilatelar Slovakia-Serbia Project. The work of the first author was supported by grants APVV-0375-06 and VEGA1/4209/07. The second author would like to thank for the support in part by the project MNTRS 144012, Grant of MTA of HTMT, French-Serbian Project”Pavle Savić”, and by the Project ”Mathematical Models for Decision Making under Uncertain Conditions and Their Applications” of the Academy of Sciences and Arts of Vojvodina supported by the

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