Formal Definition
A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (the infimum, also called the meet) and a least upper bound (the supremum, also called the join) in (L, ≤).
The meet is denoted by, and the join by .
Note that in the special case where A is the empty set the meet of A will be the greatest element of L. Likewise, the join of the empty set yields the least element. Since the definition also assures the existence of binary meets and joins, complete lattices do thus form a special class of bounded lattices.
More implications of the above definition are discussed in the article on completeness properties in order theory.
Read more about this topic: Complete Lattice
Famous quotes containing the words formal and/or definition:
“Then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part.”
—William Shakespeare (1564–1616)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)