Malaya J. Mat. 2(3)(2014) 222–227 Lattice for covering rough approximations Dipankar Rana a, ∗ and Sankar Kumar Roy b a,b Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India. Abstract Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this type of data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough sets. This paper presents the lattice based on covering rough approximations and lattice for covering numbers. An important result is investigated to illustrate the paper. Keywords: Covering, Rough Set, Lattice, Covering approximation. 2010 MSC: 03G10, 14E20, 18B35. c 2012 MJM. All rights reserved. 1 Introduction Theory of rough sets was introduced by Z. Pawlak [7], assumed that sets are chosen from a universe U, but that elements of U can be specified only upto an indiscernibility equivalence relation E on U. If a subset X ⊆ U contains an element indiscernible from some elements not in X, then X is rough. Also a rough set X is described by two approximations. Basically, in rough set theory, it is assume that our knowledge is restricted by an indiscernibility relation. An indiscernibility relation is an equivalence relation E such that two elements of an universe of discourse U are E-equivalent if we can not distinguish these two elements by their proper- ties known by us. By the means of an indiscernibility relation E, we can partition the elements of U into three disjoint classes respect to any set X ⊆ U, defined as follows: • The elements which are certainly in X. These are elements x ∈ U whose E-class x/E is included in X. • The elements which certainly are not in X. These are elements x ∈ U such that their E-class x/E is included in X co , which is the complement of X • The elements which are possibly belongs to X. These are elements whose E-class intersects with both X and X co . In other words, x/E is not included in X nor in X co . From this observation, Pawlak [7] defined lower approximation set X ↓ of X to be the set of those elements x ∈ U whose E-class is included in X, i.e, X ↓= {x ∈ U : x/E ⊆ X} and for the upper approximation set X ↑ of X consists of elements x ∈ U whose E-class intersect with X, i.e, X ↑= {x ∈ U : x/E ∩ X  = ∅}. The difference between X ↓ and X ↑ is treated as the actual area of uncertainty. Covering-based rough set theory ([17], [19]) is a generalization of rough set theory. The structure of covering- based rough sets ([18],[19],[20]) have been a interested field of study. The classical rough set theory is based on equivalence relations. An equivalence relation corresponds to a partition, while a covering is an extension of a partition. This paper focuses on establishing algebraic structure of covering-based rough sets through down-sets and up-sets. Firstly, we connect posets with covering-based rough sets, then covering-based rough sets can be investigated in posets. Down-sets and up-sets are defined in the poset environment. In order to ∗ Corresponding author. E-mail address: sankroy2006@gmail.com (Sankar Kumar Roy).