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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 [Formula: see text] be a Type-1 (Type-2) bi-base of ordered b-semiring and [Formula: see text] such that [Formula: see text] but not [Formula: see text], nor [Formula: see text] ([Formula: see text], nor [Formula: see text]). We introduce the notions of one-quasi order and two-quasi order on an ordered [Formula: see text]-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.
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Palanikumar et al. (2022) studied this question.
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