Inclusion compatibilities are a generalization of referential constraints, and therefore provide the foundation on which referential integrity is based. In common with functional compatibilities, inclusion compatibilities represent one-to-many relationships; however, inclusion compatibilities typically represent relationships between relations (see Database-Level Constraints), while functional compatibilities 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
| PK | part_name | color | weight | city |
| part_num | ||||
| PK | quantity | |
| FK | FK | |
| supp_num | 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. This defines a simple foreign key-primary key relationship. However, to write a proper referential integrity relationship, the right hand side (RHS) can indicate any candidate key of the specified relation variable, not necessarily its primary key (see Foreign Key Constraints).
The left hand side (LHS) and RHS of a compatibility relationship are optional 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:
| Axiom | Formal Expression | |||||
|---|---|---|---|---|---|---|
| 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 |