In this note we introduced a P-Krull monoid, which is a weak but parallel to Krull monoid, we also characterized a P-Krull monoid as a BFM.

In this manuscript, we introduce and discuss the term bipolar picture fuzzy graphs along with some of its fundamental characteristics and applications. We also initiate the concepts of complete bipolar picture fuzzy graphs and strong bipolar picture fuzzy graphs. Firstly, we apply different types of operations to bipolar picture fuzzy graphs and then we introduce various products of bipolar picture fuzzy graphs. Several other terms such as order and size, path, neighbourhood degrees, busy values of vertices and edges of bipolar picture fuzzy graphs are also discussed. These terminologies also lay the foundations for the discussion about the regular bipolar picture fuzzy graphs. Moreover, we also discuss isomorphisms, weak and co-weak isomorphisms and automorphisms of bipolar picture fuzzy graphs. Finally, at the base of bipolar picture fuzzy graph we present the construction of a bipolar picture fuzzy acquaintanceship graph, which would be an important tool to measure the symmetry or asymmetry of acquaintanceship levels of social networks, computer networks etc.

No abstract

Theory of the Cayley graphs is directly linked with the group theory. However, if there are uncertainties on the vertices or edges or both then fuzzy graphs have an extraordinary importance. In this perspective, numbers of generalizations of fuzzy graphs have been explored in the literature. Among the others, picture fuzzy graph (PFG) has its own importance. A picture fuzzy graph (PFG) is a pair G ¼ ðC; DÞ defined on a H Ã = ðA; BÞ, where C = ð C ; C ; # C Þ is a picture fuzzy set on A and D = ð D ; D ; # D Þ is a picture fuzzy set over the set B A Â A such that for any edge mn 2 B with D ðm; nÞ minð C ðmÞ; C ðnÞÞ, D ðm; nÞ minð C ðmÞ; C ðnÞÞ and # D ðm; nÞ ! maxð# C ðmÞ; # C ðnÞÞ: In this manuscript, we introduce the notion of the Cayley picture fuzzy graphs on groups which is the generalization of the picture fuzzy graphs. Firstly, we discuss few important characteristics of the Cayley picture fuzzy graphs. We show that Cayley picture fuzzy graphs are vertex transitive and hence regular. Then, we investigate different types of Cayley graphs induced by the Cayley picture fuzzy graphs by using different types of cuts. We extensively discuss the term connectivity of the Cayley picture fuzzy graphs. Vertex connectivity and edge connectivity of the Cayley picture fuzzy graphs are also addressed. We also investigate the linkage between these two. Throughout, we provide the extensions of some characteristics of both the PFGs and Cayley fuzzy graphs in the setting of Cayley picture fuzzy graphs. Finally, we provide the model of interconnected networks based on the Cayley picture fuzzy graphs.

<abstract> <p>Our purpose in this paper is to initiate and study the notions of fuzzy subnear-semirings and fuzzy soft subnear-semirings. We study few of their elementary properties by providing suitable examples. Moreover, we present the characterizations of zero symmetric near-semirings (seminearrings) through their fuzzy ideals and fuzzy soft ideals. Fuzzy soft anti-homomorphism of fuzzy soft near-semirings and fuzzy soft R-homomorphisms of fuzzy soft R-subsemigroups are also introduced and discussed.</p> </abstract>