The pair is called a partial order, when is a partial order on *D*,
that is, a reflexive, antisymmetric and transitive relation. Let . We say that *d* is a
lower bound of *X* if, for all , . *d* is the greatest lower bound
(glb) of *X*, if *d* is a lower bound, and, for every *e* such that *e* is a lower bound of *X*, we have
. An analogue definition holds for upper bounds and the least upper bound (lub). We
say that is a minimal (or bottom) element of *D*, if is a lower bound for
*D*.

Definition 136.is a complete partial order (cpo) if is a partial order, and, for every increasing, countable sequence (so-called chain) , , of elements ofDits least upper bound exists.

Definition 138.LetD,Ebe complete partial orders. A function is continuous, if it is monotonic, and, for every chain of elements ofD,

The set of continuous functions from the cpo

Dto the cpoEwill be denoted by .

Example 139.([11])

Every set ordered by the identity function is a discrete cpo. Any function from discrete cpo’s to discrete cpo’s are continuous.

Let be the cpo consisting of elements and such that .

Let

Xbe a set. Let . Then is a complete lattice.Let . Then

Fis a cpo.Let be the partial order . Then is a cpo. Let be such that

Then is continuous for every , but is not continuous.

Definition 141.Let be a partial order, let . ThenA fixpoint of

fis both a prefixpoint and a postfixpoint.dis called the least fixpoint (lfp) off, ifdis a fixpoint, and, for every , .

Without proof we state the following theorem.

Theorem 142.(Knaster and Tarski) Let be a cpo with bottom, assume . Then

is the least fixpoint of

f, where , , and . Moreover, is the least prefixpoint off, that is, if , then .

Below, we consider a straightforward application of the Knaster-Tarski theorem. Let *X* be a set, and
be a relation over *X*. As before, let

and let . We assert the following lemma.

Proof.First of all, observe that is continuous. For, let be a chain in . Then

This means, Theorem 142 is applicable. Let , and for . Thus,

By this, the lemma is proved.

As a consequence, we prove that indeed defines the reflexive, transitive closure of
*R*.

Corollary 144.Let , and let be defined as above, and let . Furthermore, assume is such thatQis reflexive, transitive and . Then

Proof.By the Knaster–Tarski theorem it is enough to verify thatQis a prefixpoint of . By the reflexivity ofQ, we have , moreover, by assumption, , which, together with transitivity, yields .