Dependency structures for decision tables

In this paper we study decision tables from a more general and abstract outlook. We focus our attention on consistency and inconsistency of decision tables. The starting point of our analysis is the remark that an inconsistent table has different local degrees of consistency depending on how an object and a condition attribute subset are chosen. When X and A run respectively over the object and the condition attribute set, we describe the interrelations of local consistencies by means of two set operators. They enable us to generalize the classical Pawlak's attribute dependency function. The operatorial standpoint correlates the study of decision tables in RST to the classical mathematical theories investigated through functional operators. In this perspective, we are also interested in finding which condition attribute subsets preserve the local positive region. This is the main reason to introduce the notions of local positive essentials and local positive reducts. These attribute subset families, in general, do not satisfy the properties of their counterparts in information table theory. Hence, in order to extend these results to decision tables, we can follow two different approaches: to define a subclass of decision tables in which they hold or to change the nature of the hypergraph induced by the decision discernibility matrix.