| In this paper, we investigate the Lawson topology on complete semilattices and show that the Hausdorffness, monotone Hausdorffness, normality, and monotone normality are equivalent with respect to the Lawson topology on complete semilattices. On upper semi-lattices we have the same result.Moreover, We give some equivalent conditions of the continuity on strong generalized continuous complete semilattices. If a generalized continuous complete semilattice or a generalized continuous upper semilattice has a countable basis defined in 3.3.2, then the Lawson topology on them is meteizable.The last part of this paper gives some characterization of congruence R on the generalized continuous upper semilattice L and show that L/R is a generalized continuous upper semilattice iff R is a subalgebra iff there is a closure operator c on L such that the inverse image of the diagnal of L x L is R under the mapping cx c. |
