definition of set of lower bounds of subset 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 lower bounds 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, \(\lt'\)
\( S\): \(\subseteq S'\)
\(*Lb (S)\): \(= \{s' \in S' \vert \forall s \in S \setminus \{s'\} (s' \lt' s)\}\)
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Conditions:
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2: Note
\(\forall s \in S \setminus \{s'\} (s' \lt' s)\) is equivalent with \(\forall s \in S (s' \le' s)\), where \(s' \le' s\) means that \(s' \lt' s\) or \(s' = s\), because if \(s'\) satisfies the former, if \(s' \in S\), for \(s = s' \in S\), \(s = s'\) and for each \(s \in S\) such that \(s \neq s'\), \(s \in S \setminus \{s'\}\), so, \(s' \lt' s\), so, \(s'\) satisfies the latter, and if \(s' \notin S\), for each \(s \in S\), \(s \in S \setminus \{s'\}\), so, \(s' \lt' s\), so, \(s'\) satisfies the latter; if \(s'\) satisfies the latter, for each \(s \in S \setminus \{s'\}\), \(s' \le' s\), but as \(s \neq s'\), \(s' \lt' s\), so, \(s'\) satisfies the former.
\(Lb (S)\) may be empty.
When \(S = \emptyset\), \(Lb (S) = S'\), because each \(s' \in S'\) vacuously satisfies the condition.
