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تسجيل

0,1-Simple Lattice

المؤلف:  Grätzer, G

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

الجزء والصفحة:  ...

31-12-2021

1858

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 {1}" src="https://mathworld.wolfram.com/images/equations/01-SimpleLattice/Inline12.gif" style="height:16px; width:51px" /> and {0}" src="https://mathworld.wolfram.com/images/equations/01-SimpleLattice/Inline13.gif" style="height:16px; width:51px" />.

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

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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.