Abstract
In this paper, the concepts of C-precontinuous posets, quasi C-precontinuous posets and meet Cprecontinuous posets are introduced. The main results are: (1) A complete semilattice P is C-precontinuous (resp., quasi C-precontinuous) if and only if its normal completion is a C-continuous lattice (resp., quasi C-continuous lattice); (2) A poset is both quasi C-precontinuous and Frink quasicontinuous if and only if it is generalized completely continuous; (3) A complete semilattice is meet C-precontinuous if and only if its normal completion is meet C-continuous; (4) A poset is both quasi C-precontinuous and meet C-precontinuous if and only if it is C-precontinuous.
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