Indiscernibility relations on partially ordered sets

Let the pair (U, A) be an information system, where U is a collection of objects, the universe, and A is a finite set of attributes. If we consider a subset B of the set of attributes A, we can associate with B an indiscernibility relation on U, and thus a partition of the set U. Endow U with a partial order, obtaining a partially ordered set P, and consider an information system having P as universe. In this piece of work we investigate the notion of indiscernibility relation on a such information system. In particular, we introduce the notion of compatibility between an indiscernibility relation I on U and the partially ordered set P, and we establish a criterion for I to be compatible with P.