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Locally Bounded Lattice

A lattice L is locally bounded if and only if each of its finitely generated sublattices is bounded.

Every locally bounded lattice is locally subbounded, and every locally bounded lattice L has a bounded hyperfinite extension in any nonstandard enlargement ^*L. This latter nonstandard property characterizes locally subbounded lattices.

A locally bounded lattice is locally tight if and only if each of its hyperfinitely generated extensions is internally tight. One can also prove the following result, using nonstandard characterizations of these notions: Let L be a locally finite lattice with at least one strictly increasing meet endomorphism and at least one strictly decreasing join endomorphism. If L is locally tight, then it is bounded.

See also

Locally Subbounded Lattice

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

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References

Grätzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, 1971.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

Locally Bounded Lattice

Cite this as:

Insall, Matt. "Locally Bounded Lattice." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LocallyBoundedLattice.html

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