A poset is said to be continuous if for every
the set is a directed set,
In the first condition, indicates the way below relation on . It is true that in any poset, if exists, then . So for a poset to be continuous, we require that .
A continuous lattice is a complete lattice whose underlying poset is continuous. Note that if is a complete lattice, condition 1 above is automatically satisfied: suppose and with , then there are finite subsets of with and . Then is finite and , or , implying that is directed.
A chain is continuous iff it is complete.
The lattice of ideals of a ring is continuous.
Every algebraic lattice is continuous.
- 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).
|Date of creation||2013-03-22 16:43:18|
|Last modified on||2013-03-22 16:43:18|
|Last modified by||CWoo (3771)|