Criterion for equational Noetherianity and complexity of the solvability problem for systems of equations over partially ordered sets

Results are presented concerning the main problem of algebraic geometry over partially ordered sets from a computational point of view, namely, the solvability problem for systems of equations over a partial order. This problem is solvable in polynomial time if the directed graph corresponding to the partial order is a adjusted interval digraph, and is NP-complete if the base of the directed graph corresponding to the partial order is a cycle of length at least 4. We also present a result characterizing the possibility of transition from infinite systems of equations over partial orders to finite systems. Algebraic systems with this property are called equationally Noetherian. A partially ordered set is equationally Noetherian if and only if any of its upper and lower cones with base are finitely defined.