Kleene fixed-point theorem

In mathematics, the Kleene fixpoint theorem of order theory states that given any complete lattice L, and any continuous (and therefore monotone) function

${\displaystyle f:L\to L,}$

the least fixed point (lfp) of f is

${\displaystyle {\textrm {lub}}\left(\left\{f^{i}({\textrm {bot}}_{L})\mid i\in \mathbb {N} \right\}\right)}$

where ${\displaystyle {\textrm {bot}}_{L}}$ is the bottom element of ${\displaystyle L}$ and ${\displaystyle {\textrm {lub}}}$ denotes least upper bound.

See also

• Other fixed-point theorems