On the notion of N-potency in simple semigroups

Let S be a semigroup with the Kerner N. Let S′ be any subset of S, S′⊆S. We shall say that S′ is N-potent if for some integer ρ≧1, S1ρ⊆N. hold. We prove first two simple theorems. 1. The sum of a finite number of N-potent left (right) ideals is a N-potent left (right) ideal. 2. Every N-potent left (right) ideal is contained in some two-sided N-potent ideal. Now, we introduce in the sdudy of semigroups two conditions, which we shall use in the following formulation. Conditions a1 : S is a semigroup with the kernel N, having at least one N-potent simple left ideal. Conditions B1 : S is a semigroup with the kerner N having at least one N-potent simple left and at least one N-potent simple right ideal. Then, we can prove five important theorems. 1. Let s be a simple semigroup having at least one N-potent left ideal of s which itself contains N. Then, a) S2=S b) every left ideal L⊃N is N-potent 2. If s has at least one N-potent left ideal L⊃N, then S is N-potent and hense all left ideals of S are N-potent. 3. Let S be a simple semigroup satisfying Condition a1. Then, a) S is the class sum of its simple left ideals b) If L* is one fixed simple left ideal of S, then every simple left ideal of S is L=N+L*c with c∈L. 4. On S satisfying condition a1 , for every two simple left ideals Lα, Lβ of S there hold always; LαLβ=Lβ 5. Let S be a simple semigroup satisfying condition a1. Then to every a∈S-N there exists an element e∈S-N with a=ea. Moreover the following implication holds : a=αb→αa=b (a,b,α∈S-N and α∈S-N