On a Matrix over NC and Multiset NC Semigroups

Abstract

In this paper, we define a matrix over neutrosophic components (NCs), which was built using the four different intervals $\left(\mathrm{0,1}\right),\left[\left.\mathrm{0,1}\right)|,|\left(\mathrm{0,1}\right.\right]$ , and $\left[\mathrm{0,1}\right]$ . This definition was made clear by introducing some examples. Then, the study of the algebraic structure of matrices over NC under addition modulo 1, the usual product, and product by using addition modulo 1 was introduced, from which it was found that the matrix over NC built using interval $\left[\mathrm{0,1}\right)$ happens to be an abelian group under addition modulo 1. Furthermore, it is proved that the matrix over NC defined on the interval $\left[\mathrm{0,1}\right)$ is not a regular semiring. Also, we define a matrix over multiset NC semigroup using the interval $\left[\mathrm{0,1}\right)$ . Moreover, we define a matrix over m-multiplicity multiset NC semigroup for finite m. Several interesting properties are discussed for the three structures. It was concluded that the last two structures are semigroups and semirings under addition modulo 1 and usual product, respectively.