Different Partial Prime Bi-Ideals and Its Extension of Partial Ternary Semirings

We discuss the partial bi-ideal of partial ternary semirings S. The partial bi-ideal is a new generalization of the ideal. To determine the relationships between the three types of partial prime bi-ideals and their examples. We constructed the partial right ideal, partial lateral ideal, partial left ideal, partial ideal, and partial bi-ideal generated by a single element. We interact with the relationships between HQ, LQ and RQ, where Q is bi-ideal. Consequently, we defined three distinct partial m-systems. The partial bi-ideal P of S is a partial 2-prime if and only if Z1Z2Z3⊆P, where Z1 is a partial right ideal, Z2 is a partial lateral ideal and Z3 is a partial left ideal of S implies either one of Z1⊆P, Z2⊆P and Z3⊆P. Also, we discuss HQ is a unique biggest two-sided partial prime ideal of S contained Q. Suppose that M is a partial m3-system and partial bi-ideal Q of S with Q∩M is empty, there exists a partial 3-prime P of S containing Q which includes P∩M=∅. Finally, examples were provided to illustrate the results.