Realizing posets as prime spectra of Leavitt path algebras

TL;DR

This paper constructs a method to represent a broad class of partially ordered sets as the prime spectra of Leavitt path algebras by associating graphs to posets and analyzing their algebraic properties.

Contribution

It introduces a natural graph construction for any poset and characterizes which posets can be realized as prime spectra of Leavitt path algebras.

Findings

Any poset with greatest lower bounds for downward directed subsets can be realized.

Posets satisfying the descending chain condition are included.

The construction provides a new link between order theory and algebraic structures.

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.