568: For Simplicial Complex on Finite-Dimensional Real Vectors Space, Each Simplex in Complex Is Faces of Elements of Subset of Maximal Simplexes Set

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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

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of simplicial complex.
• The reader knows a definition of maximal simplex in simplicial complex.

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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.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V$: $\in \text{ the }d\{\text{ -dimensional real vectors spaces }\}$
$C$: $\in \{\text{ the simplicial complexes on }V\}$
$S$: $=\{S_j\in C|S_j\in \{\text{ the maximal simplexes in }C\}\}$, $\subseteq C$
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Statements:
$\mathrm{\forall }{S}_k\in C(\mathrm{\exists }{S}^{\prime }\subseteq S(S^{\prime }\ne \mathrm{\varnothing }\wedge \mathrm{\forall }{S}_j\in S^{\prime }(S_k\in \{\text{ the faces of }{S}_j\})\wedge \mathrm{\forall }{S}_j\in S\setminus S^{\prime }(S_k\notin \{\text{ the faces of }{S}_j\})))$
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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, $S^{\prime }\subseteq S$, such that $\mathrm{\forall }{S}_j\in S^{\prime }(S_k\in \{\text{ the faces of }{S}_j\})$ and $\mathrm{\forall }{S}_j\in S\setminus S^{\prime }(S_k\notin \{\text{ the faces of }{S}_j\})$.

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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 $S$.

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References

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