On quasi-ideals and bi-ideals in ternary semirings

Good and Hughes [9] introduced the notion of bi-ideal and Steinfeld [11, 12] introduced the notion of quasi-ideal. Sioson [10] studied some properties of quasi-ideals of ternary semigroups. In [1], Dixit and Dewan studied about the quasi-ideals and bi-ideals of ternary semigroups. Quasi-ideals are generalization of right ideals, lateral ideals, and left ideals whereas bi-ideals are generalization of quasi-ideals. In [2], we introduced the notion of ternary semiring. Some work on ternary semiring may be found in [3, 4, 8, 6, 7, 5]. Our main purpose of this note is to introduce the notions of quasi-ideal and bi-ideal in ternary semirings and study regular ternary semiring in terms of these two subsystems of ternary semirings.