A new method to evaluate regular ternary semigroups in multi-polar fuzzy environment

Theory of $m$-polar fuzzy set deals with multi-polar information. It is used when data comes from $m$ factors $\left(m\ge 2\right)$. The primary objective of this work is to explore a generalized form of $m$-polar fuzzy subsemigroups, which is $m$-polar fuzzy ternary subsemigroups. There are many algebraic structures which are not closed under binary multiplication that is a reason to study ternary operation of multiplication such as the set of negative integer is closed under the operation of ternary multiplication but not closed for the binary multiplication. This paper, presents several significant results related to the notions of $m$-polar fuzzy ternary subsemigroups, $m$-polar fuzzy ideals, $m$-polar fuzzy generalized bi-ideals, $m$-polar fuzzy bi-ideals, $m$-polar fuzzy quasi-ideals and $m$-polar fuzzy interior ideals in ternary semigroups. Also, it is proved that every $m$- polar fuzzy bi-ideal of ternary semigroup is an $m$-polar fuzzy generalized bi-ideal of ternary semigroup but converse is not true in general. Moreover, this paper characterizes regular and intra-regular ternary semigroups by the properties of $m$-polar fuzzy ideals, $m$-polar fuzzy bi-ideals.