s 2 -C-continuous Poset

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