Difference between revisions of "Greatest lower bound"
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− | Given a [[subset]] <math>S</math> in some larger [[partially ordered set]] <math>R</math>, a '''greatest lower bound''' or ''' | + | Given a [[subset]] <math>S</math> in some larger [[partially ordered set]] <math>R</math>, a '''greatest lower bound''' or '''infimum''' for <math>S</math> is an [[element]] <math>m \in R</math> such that <math>m \leq s</math> for every <math>s \in S</math> and there is no <math>M > m</math> with this same property. |
If the greatest lower bound <math>m</math> of <math>S</math> is an element of <math>S</math>, it is also the [[minimum]] of <math>S</math>. If <math>m \not\in S</math>, then <math>S</math> has no minimum. | If the greatest lower bound <math>m</math> of <math>S</math> is an element of <math>S</math>, it is also the [[minimum]] of <math>S</math>. If <math>m \not\in S</math>, then <math>S</math> has no minimum. |
Latest revision as of 13:55, 5 March 2022
Given a subset in some larger partially ordered set
, a greatest lower bound or infimum for
is an element
such that
for every
and there is no
with this same property.
If the greatest lower bound of
is an element of
, it is also the minimum of
. If
, then
has no minimum.
See also
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