Tight Lattice

Let  be a nontrivial bounded lattice (or a complemented lattice, etc.). Then  is a tight lattice if every proper tolerance  of  satisfies

Tight lattices play an important role in the study of congruence lattices on finite algebras. One can show that a finite lattice  is tight if and only if it is -simple and every strictly increasing meet endomorphism of  is constant. One can also show that a finite lattice  is tight if and only if its only connected tolerance is the all relation, .

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.