definition of supremum of subset of partially-ordered set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of set of upper bounds of subset of partially-ordered set.
- The reader knows a definition of subset of partially-ordered set with induced partial ordering.
- The reader knows a definition of minimum of partially-ordered set.
Target Context
- The reader will have a definition of supremum of subset of partially-ordered set.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( S'\): \(\in \{\text{ the partially-ordered sets }\}\) with any partial ordering, \(\lt'\)
\( S\): \(\subseteq S'\)
\( Ub (S)\): \(= \text{ the set of the upper bounds of } S\)
\(*Sup (S)\): \(= Min (Ub (S))\)
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Conditions:
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2: Note
\(Min (Ub (S))\) makes sense, because \(Ub (S)\) has the induced partial ordering.
This definition is not claiming that \(Sup (S)\) inevitably exists but is saying that when \(Sup (S)\) exists, it is called "supremum of \(S\)".
