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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References

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

Referenced on Wolfram|Alpha

Complete Lattice

Cite this as:

Uhl, Roland. "Complete Lattice." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CompleteLattice.html

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