0,1-Simple Lattice

Let  be a nontrivial bounded lattice (or a nontrivial complemented lattice, etc.). If every nonconstant lattice homomorphism defined on  is -separating, then  is a -simple lattice.

One can show that the following are equivalent for a nontrivial bounded lattice :

1. The lattice  is -simple;

2. There is a largest nontrivial congruence  of , and  satisfies both  and .

This result is useful in the study of congruence lattices of finite algebras.

REFERENCES:

Grätzer, G. General Lattice Theory, 2nd ed. Boston, MA: Birkhäuser, 1998.

Hobby, D. and McKenzie, R. The Structure of Finite Algebras. Providence, RI: Amer. Math. Soc., 1988.

Insall, M. "Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods." J. Austral. Math. Soc. 53, 266-280, 1992.