International Journal of Neutrosophic Science (IJNS) Vol. 1, No. 1, PP. 29-39, 2020 10.5281/zenodo.3679495 29 MBJ-neutrosophic T-ideal on B-algebra Mohsin Khalid, Neha Andaleeb Khalid*, Rakib Iqbal* Department of Mathematics, The University of Lahore, 1Km Raiwind Road, Lahore, 54000, Pakistan Department of Mathematics, Lahore Collage for Women University, Lahore, Pakistan University of Management and Technology (UMT), C-II, Johar Town, Lahore, 54000, Pakistan mk4605107@gmail.com nehakhalid97@gmail.com rakibiqbal2012@gmail.com Abstract In this paper we define and study the MBJ-neutrosophic T-ideal through diferent concept like union, intersection. further we use the important properties to investigate the MBJ-neutrosophic T-ideal under cartesian product and homomorphic results. Keywords: B-algebra, MBJ-neutrosophic set, MBJ-neutrosophic T-ideal, Cartesian product, Homomorphism. 1.Introduction [1] presented the idea of fuzzy set. [2] gave the idea of BCK-algebra. [3] introduced the BCI-algebra and it is obvius that BCK-algebra is a proper sub class of BCI-algebra. [4] wrote on intuitionistic fuzzy G-algebra. [5] introduced the intuitionistic fuzzy sets. [6] provided the idea of interval-valued intuitionistic fuzzy sets. [7] studied the neutrosophic soft subalgebra and investigted many results. [8] initiated the intuitionistic fuzzy structure of B- algebra. [9] discussed the fuzzy dot subalgebra and fuzzy dot ideal of B-algebras. [10] introduced the B-algebra in 2002. [11,12] worked on t-intuitionistic fuzzy sets in fuzzy subgroups and subrings. Barbhuiya [13] studied the t- intuitionistic fuzzy BG-subalgebra. [14] studied ideals with its level subsets. [15] introduced the T-neutrosophic cubic set and deeply investigated this set with significant characterisitics. [16] defined and studied the MBJ- neutrosophic set through subalgebra and S-extension. [17,18] The concept of Neutrosophic set is given by Smarandache with truth, indeterminate and false membership function. This paper is presented to studied the idea of MBJ-neutrosophic set throught the concept of T-ideal, homomorphic characteristics and cartesian product. 2. Preliminaries First we cite some definitions which are used to present this paper. [3] An algebra (,∗ ,0) of type (2,0) is called a BCI-algebra if it satisfies the following conditions: i) (  ∗  ) ∗ (  ∗  ) ≤ (  ∗  ),