Semicontinuous lattices (Q1272127)

Continuous lattices can be defined by means of the way-below relation ``\(\langle \langle \)'': for two elements \(x\) and \(y\) in a complete lattice \(L\), \(x\langle \langle y\) if for any ideal \(I\) of \(L\), \(y\leq \vee I\) implies \(x\in I\). Now the author introduces another relation ``\(\Leftarrow\)'' as follows: for two elements \(x\) and \(y\) in a complete lattice \(L\), \(x\Leftarrow y\) if for any semiprime ideal \(I\) of \(L\), \(y\leq \vee I\) implies \(x\in I\). The new relation is used to define semicontinuous lattices. It is shown that the main merit of this weaker form of below relation is in dealing with aspects of lattices concerning prime or pseudo-prime elements.