Algebraic structures gained from rough approximation in incomplete information systems

Abstract

We give an algebraic approach for defining rough sets on incomplete information systems. The constructed approximation sets are based on objects. Given several attributes, the value of each attribute can be known or unknown for each object. In the current paper, we introduce four different approaches, a real value, a binary, a ternary and a likelihood approach. Furthermore, we define operations on the elements of the introduced approximation sets. For all three cases we can show that the achieved structure is a quasi-Brouwer-Zadeh distributive lattice with the defined operations. We also show that the introduced lower and upper approximations build up commutative monoids with the introduced operations.