Definition:Supremum
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[edit] Ordered Set
Let
be a poset.
Let
.
An element
is the supremum of
in
if:
-
is an upper bound of
in
;
-
for all upper bounds
of
in
.
Plural: Suprema.
The supremum of
is denoted
.
The supremum of
is denoted
.
If there exists a supremum of
(in
), we say that
admits a supremum (in
).
The supremum of
is often called the least upper bound of
and denoted
.
[edit] Mapping
Let
be a mapping defined on a poset
.
Let
be bounded above on
.
It follows from the Continuum Property that the codomain of
has a supremum on
.
Thus:
.
[edit] Also see

