As shown in Table 38.2, a btree operator class must provide five comparison operators, <, <=, =, >= and >. One might expect that <> should also be part of the operator class, but it is not, because it would almost never be useful to use a <> WHERE clause in an index search. (For some purposes, the planner treats <> as associated with a btree operator class; but it finds that operator via the = operator's negator link, rather than from pg_amop.)
When several data types share near-identical sorting semantics, their operator classes can be grouped into an operator family. Doing so is advantageous because it allows the planner to make deductions about cross-type comparisons. Each operator class within the family should contain the single-type operators (and associated support functions) for its input data type, while cross-type comparison operators and support functions are “loose” in the family. It is recommendable that a complete set of cross-type operators be included in the family, thus ensuring that the planner can represent any comparison conditions that it deduces from transitivity.
There are some basic assumptions that a btree operator family must satisfy:
An = operator must be an equivalence relation; that is, for all non-null values A, B, C of the data type:
A = A is true (reflexive law)
if A = B, then B = A (symmetric law)
if A = B and B = C, then A = C (transitive law)
A < operator must be a strong ordering relation; that is, for all non-null values A, B, C:
A < A is false (irreflexive law)
if A < B and B < C, then A < C (transitive law)
Furthermore, the ordering is total; that is, for all non-null values A, B:
exactly one of A < B, A = B, and B < A is true (trichotomy law)
(The trichotomy law justifies the definition of the comparison support function, of course.)
The other three operators are defined in terms of = and < in the obvious way, and must act consistently with them.
For an operator family supporting multiple data types, the above laws must hold when A, B, C are taken from any data types in the family. The transitive laws are the trickiest to ensure, as in cross-type situations they represent statements that the behaviors of two or three different operators are consistent. As an example, it would not work to put float8 and numeric into the same operator family, at least not with the current semantics that numeric values are converted to float8 for comparison to a float8. Because of the limited accuracy of float8, this means there are distinct numeric values that will compare equal to the same float8 value, and thus the transitive law would fail.
Another requirement for a multiple-data-type family is that any implicit or binary-coercion casts that are defined between data types included in the operator family must not change the associated sort ordering.
It should be fairly clear why a btree index requires these laws to hold within a single data type: without them there is no ordering to arrange the keys with. Also, index searches using a comparison key of a different data type require comparisons to behave sanely across two data types. The extensions to three or more data types within a family are not strictly required by the btree index mechanism itself, but the planner relies on them for optimization purposes.