Quotient set
A quotient set is a set derived from another by an equivalence relation.
Let be a set, and let
be an equivalence relation. The set of equivalence classes of
with respect to
is called the quotient of
by
, and is denoted
.
A subset of
is said to be saturated with respect to
if for all
,
and
imply
. Equivalently,
is saturated if it is the union of a family of equivalence classes with respect to
. The saturation of
with respect to
is the least saturated subset
of
that contains
.
Compatible relations; derived relations; quotient structure
Let be a relation, and let
be an equivalence relation. If
and
together imply
, then
is said to be compatible with
.
Let be a relation. The relation
on the elements of
, defined as
\[\exist x\in y, P(x)\] (Error compiling LaTeX. Unknown error_msg)
is called the relation derived from by passing to the quotient.
Let be a structure,
, an equivalence relation. If the equivalence classes form a structure of the same species as
under relations derived from passing to quotients,
is said to be compatible with the structure on
, and this structure on the equivalence classes of
is called the quotient structure, or the derived structure, of
.
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