Complete Lattice

A partially ordered set (or ordered set or poset for short) (L,<=) is called a complete lattice if every subset M of L has a least upper bound (supremum, supM) and a greatest lower bound (infimum, infM) in (L,<=).

Taking M=L shows that every complete lattice (L,<=) has a greatest element (maximum, maxL) and a least element (minimum, minL).

Of course, every complete lattice is a lattice. Moreover, every lattice (L,<=) with a finite set L!=emptyset is a complete lattice.

See also

Lattice, Partially Ordered Set, Tarski's Fixed Point Theorem

This entry contributed by Roland Uhl

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Birkhoff, G. Lattice Theory, 3rd ed. Providence, RI: Amer. Math. Soc., 1967.Grätzer, G. General Lattice Theory, 2nd ed. Boston, MA: Birkhäuser, 1998.

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Complete Lattice

Cite this as:

Uhl, Roland. "Complete Lattice." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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