Realizing posets as prime spectra of Leavitt path algebras

Gene Abrams; Gonzalo Aranda Pino; Zachary Mesyan; Christopher Smith

doi:10.1016/j.jalgebra.2016.12.015

Realizing posets as prime spectra of Leavitt path algebras

Rings and Algebras 2017-02-21 v2

Authors: Gene Abrams , Gonzalo Aranda Pino , Zachary Mesyan , Christopher Smith

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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.

Keywords

partially ordered sets prime spectra Leavitt path algebras directed graphs downward directed subsets greatest lower bounds descending chain condition

Cite

@article{arxiv.1512.06771,
  title  = {Realizing posets as prime spectra of Leavitt path algebras},
  author = {Gene Abrams and Gonzalo Aranda Pino and Zachary Mesyan and Christopher Smith},
  journal= {arXiv preprint arXiv:1512.06771},
  year   = {2017}
}

Comments

27 pages. In the second version an appendix with new set-theoretic results has been added, more references have been included, several minor errors and misprints have been corrected, and the title has been changed

Realizing posets as prime spectra of Leavitt path algebras

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