In lattice theory, a bounded lattice *L* is called a **0,1-simple lattice** if nonconstant lattice homomorphisms of *L* preserve the identity of its top and bottom elements. That is, if *L* is 0,1-simple and ƒ is a function from *L* to some other lattice that preserves joins and meets and does not map every element of *L* to a single element of the image, then it must be the case that ƒ^{−1}(ƒ(0)) = {0} and ƒ^{−1}(ƒ(1)) = {1}.

For instance, let *L _{n}* be a lattice with

*n*atoms

*a*

_{1},

*a*

_{2}, ...,

*a*

_{n}, top and bottom elements 1 and 0, and no other elements. Then for

*n*≥ 3,

*L*is 0,1-simple. However, for

_{n}*n*= 2, the function ƒ that maps 0 and

*a*

_{1}to 0 and that maps

*a*

_{2}and 1 to 1 is a homomorphism, showing that

*L*

_{2}is not 0,1-simple.

## References

0,1-simple lattice Wikipedia(Text) CC BY-SA