As shown in Table 39.3, 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.