559: Maximal Simplex in Simplicial Complex

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definition of maximal simplex in simplicial complex

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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: Note

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Starting Context

• The reader knows a definition of simplicial complex.
• The reader knows a definition of face of affine simplex.

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Target Context

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

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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:
$V$: $\in \{\text{ the real vectors spaces }\}$
$C$: $\in \{\text{ the simplicial complexes on }V\}$
$\ast S_{\alpha }$: $\in C$
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Conditions:
$\mathrm{\neg }\mathrm{\exists }{S}_{\beta }\in C(S_{\alpha }\in \{\text{ the proper faces of }{S}_{\beta }\})$
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2: Natural Language Description

For any real vectors space, $V$, and any simplicial complex, $C$, on $V$, any simplex, $S_{\alpha }\in C$, such that there is no simplex, $S_{\beta }\in C$, such that $S_{\alpha }$ is a proper face of $S_{\beta }$

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3: Note

There can be multiple maximal simplexes in $C$.

This does not seem any prevalent term: as the author did not see any corresponding term in some textbooks but the author uses the concept, the term is defined here.

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References

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