Presently, interval-valued neutrosophic set theory has become an important research topic. It is widely used in various pure as well as applied fields. This chapter will provide some essential scopes to study interval-valued neutrosophic subgroup. Here the notion of interval-valued triple T-norm has been introduced, and based on that, interval-valued neutrosophic subgroup has been defined. Furthermore, some homomorphic characteristics of this notion have been studied. Additionally, based on interval-valued triple T-norm, interval-valued neutrosophic normal subgroup has been introduced and some of its homomorphic characteristics have been analyzed.
Top1. Introduction
In our real physical world, many uncertainties are involved. To tackle these ambiguities, crisp set (CS) theory is not always enough. As a result, researchers needed more capable set theories. Consequently, different set theories have emerged, for instance, fuzzy set (FS) (Zadeh, Fuzzy sets, 1965), intuitionistic fuzzy set (IFS) (Atanassov, 1986), neutrosophic set (NS) (Smarandache, 1999), plithogenic set (PS) (Smarandache, 2018), etc. FS theory is capable of handling real-life uncertainties very well. Still, in some ambiguous situations, researchers need sets that are more general i.e. IFSs or sometimes even more general sets like NSs or PSs, etc. Presently, NS theory has grabbed quite lot attentions of different researchers from various fields. Presently, NS theory has become an important and fruitful research field. Furthermore, Smarandache has also developed neutrosophic measure and probability (Smarandache, 2013), calculus (Smarandache & Khalid, 2015), psychology (Smarandache, 2018), etc. At present, NS theory is used in different applied fields, for instance, in pattern recognition problem (Vlachos & Sergiadis, 2007), image segmentation (Guo & Cheng, 2009), decision making problem (Majumdar, 2015; Abdel-Basset et. al., 2017; Abdel-Basset et. al., 2019), mobile-edge computing (Abdel-Basset et. al., 2019), neutrosophic forecasting (Abdel-Basset et. al., 2019), supply chain management (Abdel-Basset et. al., 2019; Abdel-Basset et. al., 2019), supplier selection problems (Abdel-Basset et. al., 2018; Abdel-Basset et. al., 2018), goal programming problem (Abdel-Basset et. al., 2016), multi-objective programming problem (Hezam et. al., 2015), medical diagnosis (Kumar et. al., 2015; Deli et. al., 2015), shortest path problem (Kumar, et al., 2019; Kumar et. al., 2018; Kumar et. al., 2020), transportation problem (Kumar et. al., 2019) etc. Again, gradually some other set theories, like, interval-valued FS (IVFS) (Zadeh, 1975), interval-valued IFS (IVIFS) (Atanassov, 1999) and interval-valued NS (IVNS) (Wang et. al., 2005), etc. have evolved. These notions are generalizations of CS, FS, IFS, and NSs. Presently; these set theories are extensively applied in different fields, mainly in decision-making problems (DMP). In the following Table 1 some applications of IVNSs have been discussed.
Table 1. Some applications of IVNS in various fields
| Author & Year | Applications of IVNS in various fields |
| (Broumi et. al., 2015) | Introduced the concept of n-valued IVNS and mentioned how it can be applied in medical diagnosing. |
| (Broumi et. al., 2014) | Presented the definition of parameterized soft set in IVNS environment and its application in DMPs. |
| (Ye, 2014) | Defined Hamming and Euclidean distances between two IVNSs and introduced similarity measures in IVNSs with an application in DMP. |
| (Ye, 2014) | Introduced a correlation coefficient (improved) of single-valued NSs and extended it to a correlation coefficient between IVNSs. Further, applied it in multiple attribute DMPs. |
| (Zhang et. al., 2014) | Proposed a technique based on IVNS to solve multi-criteria DMPs. |
| (Aiwu et. al., 2015) | Proposed an aggregation operation rules (improved) for IVNS and extended the generalized weighted aggregation operator. |
| (Zhang et. al., 2016) | Illustrated a novel outranking method for multi-criteria DMPs with IVNSs. |
| (Broumi et. al., 2016) | Extended the notion of neutrosophic graph-based multi-criteria decision-making approach in interval-valued neutrosophic graph theory. |
| (Deli, 2017) | Proposed the concept of the soft IVNS and investigated its application in DMP. |
| (Yuan et. al., 2019) | Applied IVNSs in image segmentation. |
| (Thong et. al., 2019) | Proposed dynamic IVNS for dynamic DMP. |