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 12:55, 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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