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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" />

Figure: r081400a

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR></table>