Accepted 17 September 2014 to the processing of distribution functions. In the framework of generalized measure theory we introduce the probabilistic-valued decomposable set functions which are related to triangle functions as natural candidates for the ‘‘addition’’ in an appropriate probabilistic pt can be found in accuracy a ormly dist tric spaces work [15], Menger proposed to replace a positive number by a distance distribution function. This fact was motiva http://dx.doi.org/10.1016/j.ins.2014.09.047 0020-0255/ 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author at: Jesenná 5, 040 01 Košice, Slovakia.

E-mail addresses: jana.borzova@student.upjs.sk (J. Borzová-Molnárová), lenka.halcinova@student.upjs.sk (L. Halcˇinová), ondrej.hutnik@upjs.sk (O. Hutník). 1 Current address: Jesenná 5, 040 01 Košice, Slovakia. 2 Current address: Jesenná 5, 040 01 Košice, Slovakia.

Information Sciences 295 (2015) 347–357

Contents lists available at ScienceDirect

Information Sciences journal homepage: www.elsevier .com/locate / insabilistic information about the (counting) measure of possibilities to win the prize. A closely related conce

Moore’s interval mathematics [16], where the use of intervals in data processing is due to measurement in to rounding. Here intervals can be considered in distribution function form linked to random variables unif over the relevant intervals. These model examples resemble the original idea of Menger of probabilistic mend due ributed . In his ted byConsider a grant agency providing a financial support for research in some area. From the set of all grant applications only the ‘‘successful’’ ones (depending on some internal rules of agency) will receive certain amount of money. So, we have only a probabilistic information about the measure of the set of ‘‘successful’’ grant applications. Of course, the knowledge of this information depends on many different aspects: the total budget of money to be divided, the internal rules of the agency, the quality of reviewers (if any), etc. Further examples are provided by lotteries, or guessing results when we have a prob-Available online 13 October 2014

Mathematics Subject Classification (2010):

Primary 54E70

Secondary 60A10, 60B05

Keywords:

Probabilistic metric space

Decomposable measure

Triangle function

Triangular norm

Aggregation function 1. Introductionmetric space. Several set functions, among them the classical (sub)measures, previously defined sT-submeasures, sL;A-submeasures as well as recently introduced Shen’s (sub)measures are described and investigated in a unified way. Basic properties and characterizations of s-decomposable (sub)measures are also studied and numerous extensions of results from the above mentioned papers are provided. 2014 Elsevier Inc. All rights reserved.Probabilistic-valued decomposable set functions with respect to triangle functions

Jana Borzová-Molnárová 1, Lenka Halcˇinová ⇑, Ondrej Hutník 2

Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice, Slovakia a r t i c l e i n f o

Article history:

Received 9 August 2013

Received in revised form 3 June 2014 a b s t r a c t

Real world applications often require dealing with the situations in which the exact numerical values of the (sub)measure of a set may not be provided, but at least some probabilistic assignment still could be done. Also, several concepts in uncertainty processing are linked thinking of situations where the exact distance between two objects may not be provided, but some probability assignment is still possible. Thus, the importance/diameter/measure of a set might be represented by a distance distribution function. cE[Fðxþ yÞP TðcEðxÞ; cFðyÞÞ; E; F 2 R; x; y > 0; ð1Þ with T being a left-continuous t-norm. Here, e0 is the distribution function of Dirac random variable concentrated at point 0. each s tem ðg fact, i additi with a number of possible generalizations based on aggregation functions in general, studying certain properties of the corwith > standing was still unclear from that paper. Thus, in this paper we provide an even deeper insight into all the mentioned

Defini s>ðG;HÞðtÞ ¼ >ðGðtÞ;HðtÞÞ; G;H 2 D ; and the ‘‘subadditivity’’ property (2) is related to the triangle inequality of the corresponding probabilistic metric space. So, 348 J. Borzová-Molnárová et al. / Information Sciences 295 (2015) 347–357we can see that triangle functions are the main ingredients connecting all the mentioned notions of (sub)measure. Thus, considering a general triangle function s on Dþ, we define and study certain properties of s-decomposable (sub)measures on a ring R of subsets of X – ; in this general setting.

In the next section, the short overview of basic notions and definitions is given. In Section 3, we introduce the basic object of our study: a s-decomposable set function with values in distance distribution functions and provide a number of concrete examples. Several properties of such set functions are then investigated in Section 4 and the results related to the probabilistic Hausdorff distance are provided in Section 6 generalizing the recent results of Shen [23]. 3 supmeasure in the terminology of Shen corresponds to submeasure in our terminology, see [23, Definition 4.1(v)].sL;TðG;HÞðxÞ ¼ sup

Lðu;vÞ¼x

TðGðuÞ;HðvÞÞ; G;H 2 Dþ; ð4Þ with a suitable operation L on Rþ. What is more, Shen’s considerations are related to the pointwisely defined (triangle) function þtion 2.3] are related to the (triangle) functionnotions which are special cases of a probabilistic-valued set function with respect to a triangle function. Recall that a triangle function s is a binary operation on Dþ such that the triple ðDþ; s;6Þ forms a commutative, partially ordered semigroup with neutral element e0.

More precisely, the notion of sT-submeasure is related to the triangle function s ¼ sT given by sTðG;HÞðxÞ ¼ sup uþv¼x