Realizing posets as prime spectra of Leavitt path algebras

We associate in a natural way to any partially ordered set $(P,\le )$ 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.