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0,1-Simple Lattice

Let L be a nontrivial bounded lattice (or a nontrivial complemented lattice, etc.). If every nonconstant lattice homomorphism defined on L is 0,1-separating, then L is a 0,1-simple lattice.

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

1. The lattice L is 0,1-simple;

2. There is a largest nontrivial congruence theta of L, and theta satisfies both [1]_theta={1} and [0]_theta={0}.

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

This entry contributed by Matt Insall (author's link)

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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.

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0,1-Simple Lattice

Cite this as:

Insall, Matt. "0,1-Simple Lattice." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/01-SimpleLattice.html

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