On the Direct Product of Intuitionistic Fuzzy Topological d-algebra

We applied the direct product concept on the notation of intuitionistic fuzzy semi d-ideals of d-algebra with the investigation of some theorems. Also, we studied the notation of direct product of intuitionistic fuzzy topological d-algebra, with the notation of relatively intuitionistic continuous mapping, on the direct product of intuitionistic fuzzy topological d-algebra.


Introduction
A d-algebra is the classes of abstract algebra introduced by Negger and Kim [1] as a useful generalization of BCK-algebra. While the idea of fuzzy set, introduced by Zadeh [2] and Atanassov [3] generalized it to the concept of intuitionistic fuzzy set. Jun et al. [4] applied this notion on dalgebra. In another line, Abdullah and Hassan [5] studied the concept of semi d -ideal on d-algebra. After that, Hasan [6] introduced the concept of intuitionistic fuzzy semi d-ideals. Here, we applied the direct product concept on the notation of intuitionistic fuzzy semi d-ideals of d-algebra, with several interesting results. We also studied the notation of the direct product of intuitionistic fuzzy topological d-algebra.

Preliminaries
We will offer here some basic concepts which we need for this study. Definition (2.1): [1] A d-algebra is a non-empty set with a constant and a binary operation with the conditions below: i. ii. iii. and , which implies that , such that . We will refer to by and iff .

Definition (2.2) : [5]
We define the semi d-ideal of as a subset of with : I) implies , II) and implies ,
We will use * + instead of * ( ) ( ) + and call it IFS for short.

Direct product of IFS d-ideal
We apply here the notation of direct product for intuitionistic set on intuitionistic fuzzy d-algebra and intuitionistic semi d-ideal. Proposition (3.1) : Let and are of , then is of . proof: We know that for any ( ) ( ) , we have The proof is completed.
We can obtain this directly from lemma 3.4 and theorem 3.5.     is an in , then is an of .

Direct product of Intuitionistic fuzzy topological d-algebra
In this section, we apply the concept of direct product for intuitionistic set on the notation of intuitionistic fuzzy topological d-algebra with some theorems of continues maps.

Conclusions
We showed in this paper that the definition of relatively intuitionistic fuzzy continuous has led us to define the notation of the direct product of intuitionistic fuzzy topological d-algebra. We also found that the homomorphism map provides the notion that the primage for the direct product of intuitionistic fuzzy topological d-algebra is also a direct product of intuitionistic fuzzy topological dalgebra. Also, the image for the direct product of intuitionistic fuzzy topological d-algebra is a direct product of intuitionistic fuzzy topological d-algebra. We believe that this work can enhance further studies in this field for the generation of direct products of finite and infinite intuitionistic fuzzy semi d-ideals on d-algebra as well as intuitionistic topological d-algebra. We hope that this work can impact upcoming research in this field or in other algebraic structures.