The notion BCI-Algebras introduced by Iséki [1], Iséki and Tanaka [2] and Iséki and Tanaka [3]. Iseki [1] introduced the notion of BCI-algebra which is a generalization of BCK-algeabra. Many mathematicians have studied the algebraic properties of BCK/BCI-algebras with many structures. There is a great ideal of literature whichhas been produced on the theory of BCK/BCI-algebras. In particular, emphasis seems to have been put on the ideal theory of BCK/BCI-algeabras. The fuzzy set initiated by Zadeh [4]. Since then this concept has been applied in many distinct branches. There are several Kinds of fuzzy sets extensions in the fuzzy set theory, like that ,intuitionistic fuzzy sets, interval valued fuzzy sets, vague sets etc. the concept of the bipolar valued fuzzy sets introduced by Lee [5] and Lee [6] as the extension of fuzzy sets. Bipolar valued fuzzy sets are seen as an extension of fuzzy sets whose membership degree range is enlarged from the interval [0,1] to [-1,1]. Xi [7] constructed the concept of fuzzy sets related to BCK, BCI, MV-algeabras. Meng and Jun [8] introduced the medial BCI-algeras. Mostafa [9] introduced the notion of medial ideals in BCI-algebras. The notion of cubic sub-algebras in BCK/BCI-algeabra initiated by Jun et al. [10], Jun and Song [11], Jun et al. [12] and Jun et al. [13]. And then several properties are investigated. Relations between a cubic sub-algebra and a cubic ideal are proved. Also, they provided characterizations of a cubic sub-algebra/ideal and discussed a method to produce new cubic sub algebra from old one. Recently Jun et al. [14] studied the notion of negative-valued function and discussed ℵ-structures. They applied ℵ-structures to BCK/BCI-algeabras and considered ℵ-subalgebras and ℵ-ideals in BCK/BCI-algeabras. Jun et al. [15] and Jun et al. [16] established an extension of a bipolar-valued fuzzy set, which introduced by Lee [5] and Lee [17]. They called it a crossing cubic structure and a lot of properties are investigate. Senepati and Yager introduced the Fermatean Fuzzy Set (FFS). Finally the paper aims to introduce the notion of fermatean crossing cubic medial ideals in BCI-algebras and several theorems and basic properties are investigated.

Preliminaries

In this section, some elementary aspects necessary for this paper are included.

During this work, we present the crossing cubic medial-ideal in BCI-algebras and considered some results of crossing crossing cubic medial ideals in BCI-algebras, the image and the pre-image of crossing cubic medial-ideal in BCI-algebra under homomorphism are defined. How the image and the pre-image of crossing cubic medial-ideal in BCI-algebra become crossing cubic medial-ideal are studied. Moreover, the product of crossing cubic medial-ideal to product crossing cubic medial-ideal is established. 

Furthermore, we construct some algorithms theory applied to medial-ideal in BCI-algebras The main purpose of our future work is to investigate the foldedness of other types of crossing cubic hyper ideals with special properties such as a crossing cubic hyper PU-ideal in hyper PU-algebras.

Acknowledgment

The authors are greatly appreciate the referees for their valuable comments and suggestion for improving the paper.

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