Order, lattices, ordered algebraic structures Lattices: basic notions Theorem

Davis's converse of the Knaster–Tarski theorem

Builds on Lattice Complete lattice Monotone function Knaster–Tarski theorem

Statement

Let $L$ be a lattice. Then $L$ is a complete lattice if and only if every monotone function $f \colon L \to L$ has a fixed point, that is, an element $x \in L$ with $f(x) = x$ .

Remarks

One direction is the Knaster–Tarski theorem: in a complete lattice, every monotone map has a fixed point, and even a complete lattice of fixed points. For the converse, if $L$ is not complete, Davis constructs a monotone map from $L$ to $L$ without fixed points. The lattice $\Z$ with its usual order is not complete, and the map $n \mapsto n + 1$ is monotone without fixed points. The result was published by Anne C. Davis in 1955.

Going deeper

For a student

The result gives a test for completeness that does not mention suprema at all. To show that a lattice is not complete, it is enough to exhibit one monotone map without a fixed point. A shift $n \mapsto n + 1$ on $\Z$ is such a map.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Complete lattice concept
Infimum concept
Lattice concept
Least fixed point concept
Monotone function concept
Supremum concept
Knaster–Tarski theorem statement
Davis's converse of the Knaster–Tarski theorem statement

Tag	`G20G1ECN56`
URI	`m2://G20G1ECN56`
Kind	statement, theorem
Writing	W0 stub
MSC	06B23, 47H10
Status	This copy of the site has no build data. Run `m2 status G20G1ECN56`.
About	Complete lattice
Uses	Lattice, Complete lattice, Monotone function, Knaster–Tarski theorem
Changed	2 November 2026