On the structure of indiscernibility relations compatible with a partially ordered set

In a recently published work the author investigates indiscernibility relations on information systems with a partially ordered universe. Specifically, he introduces a notion of compatibility between the (partially ordered) universe and an indiscernibility relation on its support, and establishes a criterion for compatibility. In this paper we make a first step in the direction of investigating the structure of all the indiscernibility relations which satisfy such a compatibility criterion