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The most simplified axiom systems of pseudo-weak- R   0  algebras and pseudo- R   0  algebras are obtained, and the mutually independence of axioms is proved. We introduce the notions of filters and normal filters in pseudo-weak- R   0  algebras. The structures and properties of the generated filters and generated normal filters in pseudo-weak- R   0  algebras are obtained. These can be seen as noncommutative generalizations of the corresponding ones in weak- R   0  algebras.

The Axioms Independence of Pseudo-Weak-R0Algebras and Filters