definition of set of minimal elements of partially-ordered set
Topics
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 set of minimal elements 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:
\( \langle S, R \rangle\): \(\in \{\text{ the partially-ordered sets }\}\)
\(*Mim (S)\): \(= \{s \in S \vert \lnot \exists s' \in S (s' R s)\}\)
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Conditions:
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2: Note
\(\lnot \exists s' \in S (s' R s)\) is different from \(\forall s' \in S \setminus \{s\} (s R s')\), because for some \(s, s' \in S\), neither \(s R s'\) nor \(s' R s\) may hold.
\(Mim (S)\) may be empty.
\(Mim (S)\) may have some multiple elements when \(R\) is properly partial (meaning non-linear), because some \(2\) maximal elements, \(s_1, s_2 \in S\), may be just not related.
For any linearly-ordered set, which is a kind of partially-ordered set, there can be no multiple minimal elements, because if \(s_1, s_2 \in S\) were minimal, exclusively \(s_1 R s_2\), \(s_1 = s_2\), or \(s_2 R s_1\), but the 1st and the 3rd cases are impossible, because \(s_2\) or \(s_1\) respectively would not be minimal.
