Rough Lattice: A Combination with the Lattice Theory and the Rough Set Theory

The rough set theory, introduced by Pawlak in 1982, is a formal for dealing with the uncertainties. But it cannot directly deal with the uncertainties with order structure. The lattice theory, introduced by Peirce and Schr$\ddot{o}$der towards the end of the nineteenth century, is a mathematical tool with order structure, algebraic structure and topological structure. In this paper, the rough theory is applied to the lattice theory, and the concept of the rough lattice is presented in order that a tool is presented which can deal with the uncertainties with lattice structure. For this purpose, an equivalence relation on a lattice is defined and then the notions of rough lattice and lower and upper approximations are introduced and some related properties are investigated. At last, some related algebraic structures are studied.