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Abstract

Let P be a finite poset.Fora subset A ofP, the open upper cover set and the open lower cover set of A is defined as U(A)= {xÎP|x covers an aÎA} and L(A)={xÎP|x is covered by an aÎA} respectively. The closed upper cover set and lower coverset of A is defined as U[A]=U(A)∪A and L[A]=L(A) ∪ A respectively. The open neighbours set of A is defined as N(A)=U(A)∪L(A). The closed neighbours set of A is defined as N[A]=N(A)∪Aand A is called a N– covering set of P if N[A] = P. The N – covering number Δ(P) is the minimum cardinality of a N-covering set. Let N_n^i be the family of all N-covering sets of a chain Pn with cardinality i and let n(Pn, i) = |N_n^i|. In this paper, we construct N_n^i, and obtain a recursive formula for n(Pn,i). Using this recursive formula we construct the polynomial N(Pn,x) = ∑_(i="" n⁄3 " " )^n▒n(Pn,i)xi called N-covering polynomial of Pn .