Loading [MathJax]/jax/output/CommonHTML/jax.js
ThmDex – An index of mathematical definitions, results, and conjectures.
Definitions
,
Results
,
Conjectures
▼
Set of symbols
▼
Alphabet
▼
Deduction system
▼
Theory
▼
Zermelo-Fraenkel set theory
▼
Set
▼
Binary cartesian set product
▼
Binary relation
▼
Binary endorelation
▼
Preordering relation
▼
Partial ordering relation
▼
Partially ordered set
▼
Minimal element
▼
Minimum element
▼
Set lower bound
▼
Set of lower bounds
Definition D301
Infimum element
Formulation 0
Let
P
=
(
X
,
⪯
)
be a
D1103: Partially ordered set
.
Let
L
B
=
L
B
P
(
E
)
be the
D553: Set of lower bounds
of
E
⊆
X
with respect to
P
.
A
D2218: Set element
x
0
∈
X
is an
infimum
of
E
with respect to
P
if and only if
(1)
x
0
∈
L
B
(2)
∀
x
∈
L
B
:
x
⪯
x
0
Formulation 1
Let
P
=
(
X
,
⪯
)
be a
D1103: Partially ordered set
.
Let
L
B
=
L
B
P
(
E
)
be the
D553: Set of lower bounds
of
E
⊆
X
with respect to
P
.
A
D2218: Set element
x
0
∈
X
is an
infimum
of
E
with respect to
P
if and only if
(1)
x
0
∈
L
B
(2)
∀
x
∈
L
B
:
(
x
,
x
0
)
∈
⪯