Some properties of nilpotent IFP semirings

We introduce and study a new class of generalized IFP semirings which is called left nilpotent IFP semirings. A semiring R is said to be left nilpotent IFP semiring if for any a \in N(R) , b \in R , ab = 0 implies aRb = 0 , where N(R) denotes the set of all nilpotent elements in R . It is shown that a left nilpotent IFP semiring need not to be IFP. Various properties of this class of semirings are studied and characterized. We also consider some kinds of extensions of left nilpotent IFP semirings. Finally, the definition of non-commutative hypergraph semigroup is given, and the hypergraph semigroups with nil-IFP are described.