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In this paper, we define the rough approximation operators in an algebra using its congruence relations and study some of their properties. Further, we consider the rough approximation operators in orthomodular lattices. We introduce the notion of rough ideal (filter) with respect to a p-ideal in an orthomodular lattice. We show that the upper approximation of an ideal J with respect to a pideal I of an orthomodular lattice is the smallest ideal containing I and J. Further we study the homomorphic images of the rough approximation operators.

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In this paper, we give a characterization theorem for rough sets based on quasi order. We obtain an algebra on the rough sets system determined by a quasi order which is the generalization of the algebra of rough sets system determined by an equivalence relation given in [1]. The properties of this algebra are abstracted at various levels and define new class of algebras. Further we give a representation theorem for the new class of algebras.

soft set theory was introduced by Molodtsov in 1999 as a mathematical tool for dealing with problems that contain uncertainty. Faruk Karaaslan et al.[6] defined the concept of soft lattices, modular soft lattices and distributive soft lattices over a collection of soft sets. In this paper, we define the concept of soft ideals and soft filters over a collection of soft sets, study their related properties and illustrate them with some examples. We also define the maximum and minimum conditions in soft lattice. In addition, we characterize soft modularity and soft distributivity of soft lattices of soft ideals.