M-systems of semirings applied to the bi-semirings via prime k-ideals
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b-semirings prime ideals k-ideals m-systemsAbstract
This paper aims to investigate certain notable classes of k-ideals of bi-semirings such as 0-prime, 1-prime, and 2-prime. Twenty sufficient and necessary conditions are established for unique type k-ideals to be type prime k-ideals in bi-semiring. A T 1 − KId (T 2 − KId) in a bi-semiring was shown to correspond to a prime T 1 − KId (prime T2 − KId) in bi-semiring whenever it is either 0-prime 1-prime
or 2-prime. There are numerous necessary and sufficient conditions for a T 1 − KId (T 2 − KId) to be a T 1 − PKId (T 2 − PKId). We show that when its complement is a τ2 2-system (τ1 2), T 1 − KId (T 2 − KId) is T 1 − PKId (T 2 − PKId). There exists a T 1 − PKId (T 2 − PKId) P such that P ∩ = ϕ and A ⊆ P for every T 1−KId (T 2 − KId) A in, where A ∩ = ϕ where is τ2 2-system (τ3 2-system).
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