Tight Lattice

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

 (0,a) in rho=>a=0, and dually (b,1) in rho=>b=1.

Tight lattices play an important role in the study of congruence lattices on finite algebras. One can show that a finite lattice L is tight if and only if it is 0,1-simple and every strictly increasing meet endomorphism of L is constant. One can also show that a finite lattice L is tight if and only if its only connected tolerance is the all relation, {(a,b)|a,b in L}.

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

Explore with Wolfram|Alpha


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.

Referenced on Wolfram|Alpha

Tight Lattice

Cite this as:

Insall, Matt. "Tight Lattice." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications