Picture fuzzy set (PFS), is a newly developed apparatus to treaty with uncertainties in problems where the opinions are yes, no, neutral, and refusal types. Extension principle is one of the key tools for describing uncertainties. It provides a general method for existing classical mathematical concepts to address fuzzy quantities. It has numerous applications in various arena of our real life. However, there are less works on extension principle for picture fuzzy sets. In this article, new extension principles namely minimal extension principle and average extension principle are proposed for picture fuzzy sets. Various properties of the minimal extension principle and the average extension principle for PFSs are also established. We also prove some properties of Zadeh’s extension principle for PFSs. Finally, arithmetic operations for PFSs based on the average extension principle are developed with numerical illustrations.

Hemirings appear in a natural manner, in some applications to the theory of automata, the theory of formal languages, graph theory, design theory and combinatorial geometry. Recently, the notions of hemirings with special structures were introduced. But still now there are no complete structural properties of hemirings. In this paper we try to investigate some structures of hemirings. This is done by introducing some examples of hemirings.

In this paper, using "belongingness) ( " and "quasi-coincidence) (q " of fuzzy points and fuzzy sets, the concept of

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