S₁-continuous?Algebraic? Partial Ordered Sets And S₁-topology

Domain theory is a perfect combination of lattice theory and topology, not only become a new direction of pure mathematics, but also has important application in theoretical computer science. Scott topology has successfully combined order structures and topological structures, already became a classical tool in research of continuous lattices. But classical Scott topology is only useful in completely lattices,which is a limitation. In the early years of 1980 s, Ern?e extended it to arbitrarily partial ordered sets, and use the cut operator, then he defined the concepts of S₁-convergence, and the topology induced by it called S₁-topology.In this paper, we continued these work, especially studied the algebraic situation and its topological properties. we have proved that an arbitrary partial ordered set is S₁-algebraic if and only if the S₁-topology on it is a strongly algebraic lattice, on the other words it is a completely distributive lattice. Then we discussed its soberity and its retration, on the end we proposed the problem of S₁-topology.