## Fixpoints

**Definition:** Given a set and a function we say that is a fixed point (fixpoint) of if .

**Definition:** Let be a partial order, let be a monotonic function on , and let the set of its fixpoints be . If the least element of exists, it is called the **least fixpoint**, if the greatest element of exists, it is called the **greatest fixpoint**.

Note:

- a function can have various number of fixpoints. Take ,
- the set of fixedpoints is
- the set of fixedpoints is
- has exactly one fixpoint:

- a function can have at most one least and at most one greatest fixpoint

We can use fixpoints to give meaning to recursive and iterative definitions

- key to describing arbitrary long executions in programs