Characterization of tri quasi purity applicable for ( b1, b2)-regularities of an ordered semirings
Keywords:
QIDs, TQIDs, ( 1, 2)-regular, tri purity, tri quasi purity.Abstract
Investigating and categorizing many different kinds of tri-ideals in ordered semirings such as ordered tri quasi ideals, quasi ideals, and (2,2)-ideals in semirings is the aim of this study. In order to characterize the strongly ( 1, 2) regular ordered semirings and fully idempotent ordered semirings. We introduce the notions of left tri purity, right tri purity, tri quasi purity, (2,2) bi purity, left weak tri purity and right weak tri purity of ordered tri ideals of an ordered semirings. Also, we discussed the 0-minimal tri-ideals, (2,1)-bi-ideals, (2,3)-bi-ideals, and (4,4)-bi-ideals in ordered semigroups. The subsemigroup in Z is a (2,2)-bi-ideals in Z if and only if there is a (2,1)-bi-ideals Ψ and a (1,2)-bi-ideals k in Z such that Ψ2 ◦k2 ⊆ ⊆ Ψ2 k2. If and only if is a tri-ideal of certain left tri-ideal in Z, then
is a (4,4)-bi-ideals in Z. We draw some inferences, even though the examples do not support the opposite.
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