Published Paper Details:

Bi-ideals constitute an important class of ideal-like structures that extend the study of one-sided and quasi-ideals in algebraic systems. In this work, we investigate the structure and characterizations of bi-ideals in ternary semirings. Beginning with the fundamental notions of sub ternary semirings, left, lateral and right ideals, regular elements, and quasi-ideals, we examine the absorption condition that determines a bi-ideal. The study establishes several structural properties of bi-ideals in ternary semirings. In particular, it is shown that every one-sided ideal and every quasi-ideal is a bi-ideal, while suitable intersections of left, lateral and right ideals also produce bi-ideals. The stability of bi-ideals under arbitrary intersections, intersections with sub ternary semirings, and suitable ternary products is established. Further, for a bi-ideal (B) and a sub ternary semiring (T), the sets (BTT) and (TTB) are shown to retain the bi-ideal structure. The study also gives structural criteria for bi-ideals arising from products (ABC) of sub ternary semirings when one of the factors is a left, lateral, or right ideal. A central characterization demonstrates that a non-empty subset (B) is a bi-ideal if and only if it can be represented as a left ideal of an appropriate right ideal, equivalently as a right ideal of an appropriate left ideal. Finally, a necessary and sufficient characterization is obtained in terms of a left ideal, a lateral ideal, and a right ideal satisfying (RILBRIL). These results provide a unified structural view of bi-ideals and clarify their relationship with the various one-sided and quasi-ideal structures in ternary semirings.