In this paper, we present one way to generalize $${\mathcal {S}}$$ S -convergence and $${\mathcal {GS}}$$ GS -convergence of nets for arbitrary posets by use of the cut operator instead of joins. Some convergence theoretical characterizations of $$s_2$$ s 2 -continuity and $$s_2$$ s 2 -quasicontinuity of posets are given. The main results are: (1) a poset P is $$s_2$$ s 2 -continuous if and only if the $${\mathcal {S}}$$ S -convergence in P is topological; (2) P is $$s_2$$ s 2 -quasicontinuous if and only if the $${\mathcal {GS}}$$ GS -convergence in P is topological.