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Date: 4-1-2022
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A partially ordered set (or ordered set or poset for short) is called a complete lattice if every subset of has a least upper bound (supremum, ) and a greatest lower bound (infimum, ) in .
Taking shows that every complete lattice has a greatest element (maximum, ) and a least element (minimum, ).
Of course, every complete lattice is a lattice. Moreover, every lattice with a finite set is a complete lattice.
REFERENCES:
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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