**1. Introduction**

It is essential for medical experts to address incomplete and uncertain information included in actual medical diagnostic questions. In order to effectively use various uncertain diagnostic information, Smarandache [1] proposed neutrosophic set (NS), which is a generalization of fuzzy set (FS) and intuitionistic fuzzy set (IFS) [2]. NS is more flexible and applicable than FS and IFS. Nevertheless, it is hard to apply the NS to practical problems for the values of the functions with respect to truth, indeterminacy and falsity lie in ]0 −,1+ [. Thus, Smarandache and Wang [3] introduced the notion of the single-valued neutrosophic set (SVNS), whose values belong to [0,1]. In the actual decision-making problems, scholars have obtained many inspiring research results according to the SVNS theories [4–9]. However, in the multicriteria decision-making problem, the application of SVNS has certain limitations. Fortunately, Yager [10] firstly discussed fuzzy multisets (FMSs), in which every element may appear more than once and may have the same or different membership values. Indeed, fuzzy multisets theories cannot cope with all types of uncertain and incomplete information. So, Ye [11]

introduced the notion of the single-valued neutrosophic multisets (SVNMSs) by capitalizing on fuzzy multisets (FMs) [12,13]. So far, a large number of scholars have studied the similarity measures of SVNMSs from different angles and discuss its application in decision-making problems in [11,14–18], which is crucial for further in-depth analysis and research on SVNMSs in the future.

As we all know, the decomposition theorem, representation theorem and expansion theorem are three theoretical pillars of fuzzy mathematics. Decomposition theorem and representation theorem are the bond between fuzzy set theory and classical set theory, that is, any fuzzy set problem can be turned into a problem of classical set by taking a cut set and constructing a geometric set. The notion of λ-cut sets of FS, some basic properties of λ-cut sets, the decomposition theorem, the representation theorem of FSs had been proposed [1,19]. What is more, the definitions of cut sets, some basic properties of cut sets, the decomposition theorem and the representation theorem of IFS, interval intuitionistic fuzzy set (IIFS), interval value fuzzy set (IVFS) which as generations of FSs had been proposed [20–28]. After that D. Singh, A. J. Alkali and A. I. Isah introduced the definition of α-cuts for FMS, which is a generalization of λ-cut sets of FS, and proposed some properties of α-cuts, decomposition theorem for FMS [29]. However, the cut sets and its operational properties, decomposition theorem and representation theorem of the SVNMSs have not been studied yet. Thus, it is necessary to discuss the cut sets, decomposition theorem and representation theorem of SVNMSs. We have already been researching SVNMSs and proposed some new results in [30–32]. Moreover, this paper proposes a new similarity from the perspective of decomposition theorem which is different from [11–16]. This new method uses the decomposition theorem as the theoretical basis and the integral as the mathematical tool. The idea is simple, the calculation is convenient, and it contains important mathematical ideas, which is more practical [33–36].

The organization of this paper is as follows: In Section 2, some basic conceptions of FMS, IFM and SVNMS are reviewed. Section 3 discusses some new properties of SVNMS. Section 4 proposes the ( α, β, γ)-cut sets for SVNMS, and investigates the decomposition theorem and the representation theorem of SVNMS. In Section 5, based on the established cut sets, a new method is proposed to calculate the similarity measure between SVNMSs. In Section 6, a practicable example is offered for medical diagnosis to illustrate the approach proposed in this paper. Section 7 presents final conclusions and further research.
