**7. Conclusions**

This paper first systematically discussed 8 properties of the union, intersection and complement of the single-valued neutrosophic multisets (SVNMSs), and showed that the complementation is no longer true in SVNMS by the counterexample. Secondly, this paper proposed the notions of cut sets and strong cut sets of SVNMSs and presented the related properties. On the basis of cut set sand strong cut sets, the decomposition theorem and representation theorem of SVNMSs were established and proved. The decomposition theorem realizes the transformation of SVNMSs and special SVNMSs. Thirdly, based on the decomposition theorem, we transformed the similarity between SVNMSs into the similarity between special SVNMSs. Therefore, we used the integral to give a new method to calculate the similarity between SVNMSs. The conceptions of new similarity measures were introduced, and its feasibility and effectiveness in multi-attribute decision making were verified accordinng to a typical example. Further, the uniqueness of the new similarity measure was analyzed by comparing the results with other similarity measures. The results obtained have a significant meaning for further theoretical research of SVNMSs. As the next research topic, we will explore the fuzzy measure and fuzzy integral of SVNMSs. In the future, we will discuss the integration of the related topics, such as neutrosophic set (multiset), fuzzy set (multiset), rough set, soft set and algebra systems (see [30–32,37–39]).

**Author Contributions:** All authors have contributed equally to this paper. The original idea of the study was proposed by X.Z., he also completed the preparation of the paper; X.Z. and Q.H conceived and designed the experiment; Q.H analyzed the experimental data and wrote the paper; The revision and submission of the paper was completed by X.Z. and Q.H.

**Funding:** This research was funded by National Natural Science Foundation of China gran<sup>t</sup> number 61573240.

**Conflicts of Interest:** The authors declare no conflict of interest.
