Inclusion compatibilities are a generalization of referential constraints. As such, they provide the foundation upon which referential integrity is based. In common with functional compatibilities (see Functional, Transitive, and Multivalued Compatibilities), inclusion compatibilities represent one-to-many relationships (see One-to-Many Relationships); however, inclusion compatibilities typically represent relationships between relations (see Database-Level Constraints), while functional compatibilities always represent relationships between the primary key of a relation variable and its attributes.
Suppose you have the following table definitions:
Using the notation R.A, where R is the name of a relation variable and A is the name of one of its attributes, you can write the following inclusion compatibility:
supplier_parts.part_num → parts.part_num
This inclusion compatibility states that the set of values appearing in the attribute part_num of relation variable supplier_parts must be a subset of the values appearing in the attribute part_num of relation variable parts. Notice that this defines a simple foreign key-primary key relationship, though it is only necessary, in order to write a proper referential integrity relationship, that the right hand side (RHS) indicates any candidate key of the specified relation variable, not necessarily its primary key (see Foreign Key Constraints for more information about this).
The left hand side (LHS) and RHS of a compatibility relationship are not required to be a foreign key and a candidate key, respectively. This is merely required to write a correct inclusion compatibility expression of a referential integrity relationship.
Inference Axioms for Inclusion Compatibilities
Interference axioms for inclusion compatibilities are described in the following table:
|Reflexive rule||A → A|
|Projection and Permutation rule||IF||AB → CD||THEN||A → C||AND||B → D|
|Transitivity rule||IF||A → B||AND||B → C||THEN||A → C|