# Analysis of Data Presented by Multisets Using a Linguistic Approach +
###### Abstract
Both classical algorithms for clustering, classification and ordering of objects, as well as algorithms specially designed to take into account the specifics of describing objects using multisets, can be applied to objects presented using the toolkit of multiset theory. The introduction of the concept and the fixation of the term'multisets' were made by [PERSON]. Then he proposed the development of ideas of multiset theory in [20]. In set theory, it is not explicitly assumed that all elements of a set are different. However, there is no fundamental prohibition on the presence of several identical elements in a set. The possibility of multiple occurrences of elements in a multiset creates a new quality that distinguishes the multiset from the usual 'ordinary' set and generates a significantly greater variety of types and features of multisets than that of sets. Multisets are sometimes referred to as bundles. Repetition sets have traditionally been studied in combinatorial mathematics [21]. The work [22] by [PERSON] analyzes the need to consider multisets as an independent mathematical object. A herewith, definitions of a multiset, union, intersection and addition of two multisets are given, some properties of these operations and examples of the use of multisets are indicated. A small summary of the basic concepts related to multisets is given in [23], where subtraction of multisets is added to the above operations. A number of properties of these operations were discussed in [24]. Later, the operations of the direct product and arithmetic multiplication of multisets, the operations of the symmetric difference of multisets, the addition and multiplication of a multiset by a number, the direct power of a multiset [25] were introduced. The concept of a fuzzy multiset was proposed by [PERSON] [24], operations on fuzzy multisets were investigated in [26; 27]. The problems of ordering multisets were studied in [28; 29]. Metric spaces of multisets and some of their properties are considered in [25; 30]. The first systematic and consistent exposition of the beginnings of multiset theory was undertaken in [31] in 2002. It introduces the main characteristics of multisets, considers possible types of multisetsand methods for their comparison, defines operations on multisets and investigates their properties, establishes rules for calculating the cardinality and dimension of an arbitrary number of multisets. In the works of [PERSON] [32], examples of the practical application of multisets for the representation of multisets are given, aspects of solving the problems of clustering, classification, and ordering of objects represented by multisets are considered. In particular, the problem of expert assessment and competitive selection of projects in a competition held in accordance with the state scientific and technical program for the study of high-temperature superconductivity is considered. Despite the insufficient'maturity' of theoretical developments [33], multisets are successfully used in various applications, in particular, in multicriteria analysis of weakly formalized problems and decision-making [25], the theory of Petri nets [23], formal language theory [34], mathematical programming [35], processing methods of heterogeneous information [24; 26; 27; 36], etc. In recent years multiset theory has been developed in the works [37; 38; 39; 40]. In [37], authors research such multiset functions as monomorphisms, epimorphisms and biomorphisms. The paper [38] discusses the problem of multiset prediction. Herewith, authors try to train a forecaster which maps an input to a multiset consisting of multiple items, and propose a novel multiset loss function by viewing this problem from the perspective of sequential decision making. In [39], authors propose a hierarchical visual architecture which is motivated by human visual attention and can be applied for multi-label image classification on a novel multiset problem with high precision and recall while localizing objects.