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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r0814001.png" /> be a [[Partially ordered set|partially ordered set]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r0814002.png" /> a [[Field|field]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r0814003.png" /> be a symbol with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r0814004.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r0814006.png" />-space is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r0814007.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r0814008.png" /> are subspaces of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r0814009.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140011.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140012.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140013.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140014.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140015.png" />-spaces; a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140016.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140017.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140020.png" />. The direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140022.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140023.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140025.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140026.png" />-space is said to be indecomposable if it cannot be written as the direct sum of two non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140028.png" />-spaces.
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The partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140029.png" /> is called subspace-finite if there are only finitely many isomorphism classes of indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140030.png" />-spaces. Kleiner's theorem asserts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140031.png" /> is subspace-finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081400/r08140032.png" /> is finite and does not contain as a full subset one of the partially ordered sets
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Let  $  S $
 +
be a [[Partially ordered set|partially ordered set]] and  $  k $
 +
a [[Field|field]]. Let  $  \omega $
 +
be a symbol with  $  \omega \notin S $.
 +
An  $  S $-
 +
space is of the form  $  V=( V _  \omega  , V _ {s} ) _ {s \in S }  $,
 +
where the  $  V _ {s} $
 +
are subspaces of the  $  k $-
 +
space  $  V _  \omega  $
 +
for  $  s \in S $,
 +
such that  $  s \leq  s  ^  \prime  $
 +
implies  $  V _ {s} \subset  V _ {s  ^  \prime  } $.
 +
Let  $  V , V  ^  \prime  $
 +
be  $  S $-
 +
spaces; a mapping  $  f: V \rightarrow V  ^  \prime  $
 +
is a  $  k $-
 +
linear mapping  $  V _  \omega  \rightarrow V _  \omega  ^  \prime  $
 +
such that  $  f( V _ {s} ) \subset  V _ {s}  ^  \prime  $
 +
for all  $  s \in S $.  
 +
The direct sum of  $  V $
 +
and  $  V  ^  \prime  $
 +
is  $  V \oplus V  ^  \prime  $
 +
with  $  ( V \oplus V  ^  \prime  ) _ {s} = V _ {s} \oplus V _ {s}  ^  \prime  $
 +
for all  $  s \in S \cup \{ \omega \} $.  
 +
An  $  S $-
 +
space is said to be indecomposable if it cannot be written as the direct sum of two non-zero  $  S $-
 +
spaces.
 +
 
 +
The partially ordered set  $  S $
 +
is called subspace-finite if there are only finitely many isomorphism classes of indecomposable $  S $-
 +
spaces. Kleiner's theorem asserts that $  S $
 +
is subspace-finite if $  S $
 +
is finite and does not contain as a full subset one of the partially ordered sets
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r081400a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r081400a.gif" />

Latest revision as of 08:11, 6 June 2020


Let $ S $ be a partially ordered set and $ k $ a field. Let $ \omega $ be a symbol with $ \omega \notin S $. An $ S $- space is of the form $ V=( V _ \omega , V _ {s} ) _ {s \in S } $, where the $ V _ {s} $ are subspaces of the $ k $- space $ V _ \omega $ for $ s \in S $, such that $ s \leq s ^ \prime $ implies $ V _ {s} \subset V _ {s ^ \prime } $. Let $ V , V ^ \prime $ be $ S $- spaces; a mapping $ f: V \rightarrow V ^ \prime $ is a $ k $- linear mapping $ V _ \omega \rightarrow V _ \omega ^ \prime $ such that $ f( V _ {s} ) \subset V _ {s} ^ \prime $ for all $ s \in S $. The direct sum of $ V $ and $ V ^ \prime $ is $ V \oplus V ^ \prime $ with $ ( V \oplus V ^ \prime ) _ {s} = V _ {s} \oplus V _ {s} ^ \prime $ for all $ s \in S \cup \{ \omega \} $. An $ S $- space is said to be indecomposable if it cannot be written as the direct sum of two non-zero $ S $- spaces.

The partially ordered set $ S $ is called subspace-finite if there are only finitely many isomorphism classes of indecomposable $ S $- spaces. Kleiner's theorem asserts that $ S $ is subspace-finite if $ S $ is finite and does not contain as a full subset one of the partially ordered sets

Figure: r081400a

see [a1]. M.M. Kleiner also has determined all the indecomposable representations of a representation-finite partially ordered set [a2]. A characterization of the tame partially ordered sets has been obtained by L.A. Nazarova [a3]. The representation theory of partially ordered sets plays a prominent role in the representation theory of finite-dimensional algebras.

References

[a1] M.M. Kleiner, "Partially ordered sets of finite type" J. Soviet Math. , 3 (1975) pp. 607–615 Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. , 28 (1972) pp. 32–41
[a2] M.M. Kleiner, "On the exact representations of partially ordered sets of finite type" J. Soviet Math. , 3 (1975) pp. 616–628 Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. , 28 (1972) pp. 42–60
[a3] L.A. Nazarova, "Partially ordered sets of infinite type" Math. USSR Izv. , 9 : 5 (1975) pp. 911–938 Izv. Akad. Nauk SSSR Ser. Mat. , 39 (1975) pp. 963–991
How to Cite This Entry:
Representation of a partially ordered set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_a_partially_ordered_set&oldid=48519
This article was adapted from an original article by C.M. Ringel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article