Interval Valued m-polar Fuzzy BCK/BCI-Algebras

Abstract

The notion of interval-valued m-polar fuzzy sets (abbreviated IVmPF) is much wider than the notion of m-polar fuzzy sets. In this paper, we apply the theory of IVmPF on BCK/BCI-algebras. We introduce the concepts of IVmPF subalgebras, IVmPF ideals and IVmPF commutative ideals and some essential properties are discussed. We characterize IVmPF subalgebras in terms of fuzzy subalgebras and subalgebras of BCK/BCI-algebras. We show that in BCK-algebra, IVmPF ideals are IVmPF subalgebras and that the converse is not valid. We provide a condition under which an IVmPF subalgebra becomes an IVmPF ideal. Further, we characterize IVmPF ideals in terms of fuzzy ideals and ideals of BCK/BCI-algebras. Moreover, we prove that in any BCKalgebra, an IVmPF commutative ideal is an IVmPF fuzzy ideal but not the converse. Also, we provide conditions under which an IVmPF ideal becomes an IVmPF commutative ideal.

Generalized Fuzzy Ideals of BCI-Algebras Based on Interval Valued m-Polar Fuzzy Structures

“…A mapping Ĝ ∶ Z → D[0, 1] is called an interval valued fuzzy set of Z, where Ĝ( ) = [G − ( ), G + ( )] for all ∈ Z , G − and G + are fuzzy sets of Z with G − ( ) ≤ G + ( ) for all ∈ Z. Definition 2.1 [30] A mapping Ĝ ∶ Z → D[0, 1] m is called an interval valued m-polar fuzzy set (briefly, IVmPF set) of Z and is defined as:…”

“…The ith projection map i is order preserving and vice versa i.e., Definition 2.2 [30] An IVmPF set Ĝ is said to be an IVmPF subalgebra if: that is, Definition 2.3 [30] An IVmPF set Ĝ is said to be an IVmPF ideal if: (1)…”

“…for all , ∈ Z and i ∈ {1, 2, … , m}. Lemma 2.4 [30] Let Ĝ be an IVmPF ideal of Z and , ∈ Z such that ≤ . Then, Lemma 2.5 [30] Let Ĝ be an IVmPF ideal of Z and , , ∈ Z such that * ≤ .…”

“…The results of BCK/BCI-algebras were developed to m-polar fuzzy set and interval valued fuzzy set frames (see [16,25,26] and more recent, [5-8, 10, 30]). Also, in [22,24,[30][31][32][33][34], some more general ideas on bipolar fuzzy were studied. In BCK/BCI-algebras and other related algebraic structures, different kinds of related concepts were investigated in various ways (see for example, [35][36][37][38][39][40][41][42][43][44][45]).…”

Regular and Intra-Regular Semigroups in Terms of m-Polar Fuzzy Environment

“…To continue their work, they introduced a new aspect of generalized m-PF ideals and studied the normalization of m-PF subalgebras in [21,22]. Recently, Muhiuddin and Al-Kadi presented interval-valued m-PF BCK/BCI-Algebras [23]. Shabir et al [24] studied regular and intra-regular semirings in terms of BFIs.…”