New approach towards different bi-base of ordered b-semiring.

The hypothesis of an ordered b-semiring provides a characterization of semiring and ordered semiring in this paper. We have attempted to investigate the Type-1 bi-ideal, Type-2 bi-ideal, Type-1 bi-base and Type-2 bi-base over ordered b-semiring in detail. Some of their characterizations are obtained through Type-1 bi-base and Type-2 bi-base. Let ℋ be a Type-1 (Type-2) bi-base of ordered b-semiring and a , b ∈ ℋ such that a ≠ b but not a ≤ t 1 b b , nor b ≤ t 1 b a (a ≤ t 2 b b , nor b ≤ t 2 b a). We introduce the notions of one-quasi order and two-quasi order on an ordered b -semiring to interact with Type-1 and Type-2 bi-bases generated by element and subset. With the help of some examples, we have shown that Type-1 bi-base and Type-2 bi-base have partial order. [ABSTRACT FROM AUTHOR]