Realizing posets as prime spectra of Leavitt path algebras
Abstract
We associate in a natural way to any partially ordered set $(P,\leq)$ a directed graph $E_P$ (where the vertices of $E_P$ correspond to the elements of $P$, and the edges of $E_P$ correspond to related pairs of elements of $P$), and then describe the prime spectrum of the resulting Leavitt path algebra $L_K(E_P)$. This construction allows us to realize a wide class of partially ordered sets as the prime spectra of rings. More specifically, any partially ordered set in which every downward directed subset has a greatest lower bound, and where these greatest lower bounds satisfy certain compatibility conditions, can be so realized. In particular, any partially ordered set satisfying the descending chain condition is in this class.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.06771
 Bibcode:
 2015arXiv151206771A
 Keywords:

 Mathematics  Rings and Algebras;
 16W10;
 16D25;
 06A06
 EPrint:
 27 pages. In the second version an appendix with new settheoretic results has been added, more references have been included, several minor errors and misprints have been corrected, and the title has been changed