is a dcpo: Suppose that is a directed subset of . Set . Define as follows: for any , where for some . Is this well-defined? Suppose . Since is directed, there is an extending both and . This means that . Therefore, is a well-defined function (on ). Hence is a dcpo.
If , then : Since extends both and , is well-defined (the construction is the same as above). To show that , suppose is directed and (note that exists by 2 above). Since , there is such that . Similarly, implies an with . Since is directed, there is with . This means and , or .
is continuous: Let . Then by 3 above, is a directed set. By 2, exists, and . Suppose . Then the function defined by is way below , for if , then for some , or , which means . Therefore, . This implies that . As a result, .
Remark. Domain theory is a branch of order theory that is used extensively in theoretical computer science. As in the example, one can think of a domain as a collection of partial pictures or pieces of partial information. Being continuous is the same as saying that any picture or piece of information can be pieced together by partial ones by way of “approximations”.
|Date of creation||2013-03-22 16:49:25|
|Last modified on||2013-03-22 16:49:25|
|Last modified by||CWoo (3771)|
|Synonym||directed complete poset|
|Synonym||directed complete partially ordered set|