View source for Representation of a partially ordered set

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.

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

<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>