definition of subset of partially-ordered set with induced partial ordering
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 subset of partially-ordered set with induced partial ordering.
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, \(\lt'\)
\(*S\): \(\subseteq S'\) with the partial ordering, \(\lt := \lt' \cap (S \times S)\)
//
Conditions:
//
2: Note
Let us see that \(\lt\) is indeed a partial ordering.
Note that any partial ordering is a relation, which is a set of some ordered pairs.
\(\lt' \subseteq S' \times S'\).
So, \(\lt' \cap (S \times S)\) makes sense, and \(\lt \subseteq S \times S\) is a relation.
Let us see that \(\lt\) satisfies the conditions to be a partial ordering.
1) \(\lt\) is irreflexive: for any element, \(s \in S\), \(\lnot s \lt s\), because \(s \in S'\), and \(\lnot s \lt' s\), which means that \((s, s) \notin \lt'\), and \((s, s) \notin \lt' \cap (S \times S) = \lt\).
2) \(\lt\) is transitive: for any elements, \(s_1, s_2, s_3 \in S\), such that \(s_1 \lt s_2\) and \(s_2 \lt s_3\), \(s_1 \lt s_3\), because \(s_1, s_2, s_3 \in S'\) and \(s_1 \lt' s_3\), which means that \((s_1, s_3) \in \lt'\), and \((s_1, s_3) \in \lt' \cap (S \times S) = \lt\).
