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

