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Difference between revisions of "Chain"

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The same as a [[Totally ordered set|totally ordered set]].
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A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021380/c0213801.png" /> is an element of the tensor product of a chain complex by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021380/c0213802.png" />.
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===In ordered sets===
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The same as a [[totally ordered set]]: in a general [[partially ordered set]], a subset which is totally ordered with respect to the induced order.  The [[rank of a partially ordered set]] is the maximal cardinality of a chain.
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===In algebraic topology===
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A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group $G$ is an element of the tensor product of a chain complex by the group $G$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,  S. Eilenberg,  "Foundations of algebraic topology" , Princeton Univ. Press  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.J. Hilton,  S. Wylie,  "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press  (1960)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,  S. Eilenberg,  "Foundations of algebraic topology" , Princeton Univ. Press  (1966) {{ZBL|0047.41402}}</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  P.J. Hilton,  S. Wylie,  "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press  (1960) {{ZBL|0091.36306}}</TD></TR>
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</table>

Latest revision as of 15:33, 18 March 2023


In ordered sets

The same as a totally ordered set: in a general partially ordered set, a subset which is totally ordered with respect to the induced order. The rank of a partially ordered set is the maximal cardinality of a chain.

In algebraic topology

A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group $G$ is an element of the tensor product of a chain complex by the group $G$.

References

[1] N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966) Zbl 0047.41402
[2] P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) Zbl 0091.36306
How to Cite This Entry:
Chain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain&oldid=12098
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article