Volume 48 | Number 4 | Year 2017 | Article Id. IJMTT-V48P535 | DOI : https://doi.org/10.14445/22315373/IJMTT-V48P535

Let P be a finite poset.Fora subset A ofP, the open lower cover set of A is defined as L(A)={xP|x is covered by an aA}. The closed lower coverset of A is defined as L[A]=L(A) โช A and A is called an L โ covering set of P if L[A] = P. The L โ covering number โ(P) is the minimum cardinality of aL-covering set. Let ๐ฟ๐ ๐ be the family of all L-covering sets of a chain Pn with cardinality i. Similarly we can define Uโcovering and N-covering sets of Pn with cardinality i. โ(Pn,i) = |๐ฟ๐ ๐ |, ๐(Pn,i) = |๐๐ ๐ |, ๐(Pn,i) = |๐๐ ๐ |. In this paper, we construct ๐ฟ๐ ๐ , and obtain a recursive formula for โ(Pn,i). Using this recursive formula we construct the polynomial L(Pn,x) = โ ๐ ๐=0 ๐ 2 (Pn,i)xi called L-covering polynomial of Pn .

[1] M. Bayer and J. Billera, Counting chains and Faces in Polytopes andPosets, Contemporary Mathematics, 34, 207-252 (1984).

[2] P. Crawley and R.P.Dilworth, Algebraic theory of Lattices, Prentice-Hall, Inc. Englewod, Cliffs, New Jersey. 1973.

[3] B.A Davey and H.A.Priestley, Introduction to Lattices and Order, Second Edition, Cambridge University Press, 2002.

[4] Garrett Birkhoff, Lattice Theory, American Mathematical Society Colloquim Publications, Vol.XXV, 1961.

[5] G.Grรคtzer, General Lattice Theory, BirkhauserVerlag, Basel, 1978.

[6] Greene C. On the Mobius algebra of a partially ordered set, Advances in math. 10, 177-187(1973).

[7] Gunter M. Ziegler, Lectures on Polytopes, Springer โ Verlag, New York, Inc., 1995.

[8] Paffenholz and Andreas, Construction for posets, Lattices, and Polytopes, Doctoral Dissertation, School of Mathematical and Natural Sciences, Technical University of Berlin, 2005.

[9] SaeidAlikhani and Yee-Hock Peng, Dominating Sets and Domination Polynomials of Paths, International Journal of Mathematics and Mathematical Sciences, Vol.2009, PP.1-10.

[10] R.P.Stanley, Enumerative Combinatorics, Volume 1, Wordsworth and Brooks / Cole, 1986.

[11] R.Subbarayan andA.Vethamanickam, On the Lattice of Convex Sub lattices, Elixir Dis. Math. 50 (2012), 10471-10474.

[12] A.Vethamanickam, Topics in Universal Algebra, Ph.D. Thesis, Madurai Kamaraj University, 1994.

[13] A.Vethamanickam,and R.Subbarayan, Simple extensions of Eulerian Lattices, Acta Math. Univ. Comenianae, Vol. LXXIX, I(2010), PP.47-54.

[14] A.Vethamanickam and R.Subbarayan, Some Properties of Eulerian Lattices, CommentationesMathematicaeUniversitatitsCarolinae, Vol.55 (2014), (4) PP.499-507.

[15] A.Vethamanickam and K.M.Thirunavukkarasu, U-covering sets and U-covering polynomials of chains, International Journal of Mathematical Archive, Vol.8 (8), 2017, 41-44.

K.M. Thirunavukkarasu, A. Vethamanickam, "L-Covering Sets and L-Covering Polynomials
of Chains," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 48, no. 4, pp. 237-239, 2017. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V48P535