s2-C-continuous Poset

In this paper, we introduce the concept of S2-C-continuous poset by cut operator. The main results are: (1) A sup-semilattice is both S2-C-continuous and S2-continuous if and only if it is S2-CD; (2) A sup-semilattice is both S2-C-continuous and hypercontinuous if and only if it is S2-CD; (3) A sup-semilattice is both S2-QC-continuous and S2-quasicontinuous if and only if it is S2-GCD; (4) A sup-semilattice is both S2-QC-continuous and quasi-hypercontinuous if and only if it is S2-GCD; (5) A poset is S2-C-continuous if and only if it is both S2-MC-continuous and S2-QC-continuous; (6) A poset is S2-CD if and only if its order dual is S2-CD; (7) A semi-lattice is S2-GCD if and only if its order dual is hypercontinuous; (8) The lattice of all σ2-closed subsets of a poset is C-continuous; (9) A poset P is S2-continuous if and only if the lattice C2(P) of all σ2-closed subsets of P is a continuous lattice if and only if C2(P) is a CD lattice; (10) A poset P is S2-quasicontinuous if and only if the lattice σ2(P) of all σ2-open subsets of P is a hypercontinuous lattice if and only if C2(P) is a GCD lattice if and only if C2(P) is a quasicontinuous lattice.