description/proof of that for simplicial complex on finite-dimensional real vectors space, each simplex in complex is faces of elements of subset of maximal simplexes set
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of simplicial complex.
- The reader knows a definition of maximal simplex in simplicial complex.
Target Context
- The reader will have a description and a proof of the proposition that for any simplicial complex on any finite-dimensional real vectors space, each simplex in the complex is the faces of the elements of a subset of the maximal simplexes 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:
\(V\): \(\in \text{ the } d\{\text{ -dimensional real vectors spaces }\}\)
\(C\): \(\in \{\text{ the simplicial complexes on } V\}\)
\(S\): \(= \{S_j \in C \vert S_j \in \{\text{ the maximal simplexes in } C\}\}\), \(\subseteq C\)
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Statements:
\(\forall S_k \in C (\exists S' \subseteq S (S' \neq \emptyset \land \forall S_j \in S' (S_k \in \{\text{ the faces of } S_j\}) \land \forall S_j \in S \setminus S' (S_k \notin \{\text{ the faces of } S_j\})))\)
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2: Natural Language Description
For any \(d\)-dimensional real vectors space, \(V\), any simplicial complex, \(C\), on \(V\), and the set of the maximal simplexes in \(C\), \(S\), for each simplex, \(S_k \in C\), there is a nonempty subset, \(S' \subseteq S\), such that \(\forall S_j \in S' (S_k \in \{\text{ the faces of } S_j\})\) and \(\forall S_j \in S \setminus S' (S_k \notin \{\text{ the faces of } S_j\})\).
3: Proof
Each simplex in \(C\) is definitely a maximal simplex or not any maximal simplex. So, \(S\) is well-defined.
Each simplex, \(S_1 \in C\), is a face of at least 1 maximal simplex, because if \(S_1\) is a maximal simplex, \(S_1\) is a face of maximal \(S_1\); if \(S_1\) is not any maximal simplex, \(S_1\) is a proper face of another simplex, \(S_2 \in C\); if \(S_2\) is a maximal simplex, \(S_1\) is a face of maximal \(S_2\); and so on; \(S_1, S_2, ...\) are all different, because \(S_1 \subset S_2 \subset ...\); note that \(S_{j + 1}\) has more vertexes than \(S_j\); as \(V\) is \(d\)-dimensional, any simplex can have at most \(d + 1\) vertexes, and the process has to stop at an \(S_n\), where \(S_n\) is a maximal simplex; so, \(S_1\) is a face of maximal \(S_n\).
While \(S_1\) is a face of at least 1 element of \(S\), \(S_1\) can be also the faces of some other elements of \(S\). Of each element of \(S\), \(S_1\) is definitely a face or not, so, \(S_1\) is the faces of the elements of the definitely-determined nonempty subset of \(S\).
