definition of minimum of partially-ordered set
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About: set
The table of contents of this article
Starting Context
- The reader knows a definition of partially-ordered set.
Target Context
- The reader will have a definition of minimum 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, \(\lt\)
\(*Min (S)\): \(\in S\)
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Conditions:
\(\forall s \in S \setminus \{Min (S)\} (Min (S) \lt s)\)
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2: Note
This definition is not claiming that such a \(Min (S)\) inevitably exists, but is saying that if such a \(Min (S)\) exists, it is called "minimum of \(S\)".
There cannot be more than \(1\) minimums, because if there were some \(2\) minimums, \(s, s'\), such that \(s \neq s'\), \(s \lt s'\) and \(s' \lt s\), which would imply that \(s \lt s\), a contradiction against the irreflexiveness.
