For a partially ordered set with a greatest element, a subset is cofinal if and only if it contains that greatest element ( this follows, since a greatest element is necessarily a maximal element ).
42.
For a partially ordered set with a greatest element, a subset is cofinal if and only if it contains that greatest element ( this follows, since a greatest element is necessarily a maximal element ).
43.
For a partially ordered set with a greatest element, a subset is cofinal if and only if it contains that greatest element ( this follows, since a greatest element is necessarily a maximal element ).
44.
It is the least one on " L ", so " P " has least and greatest elements, or more generally that every monotone function on a complete lattice has least and greatest fixpoints.
45.
Every Heyting algebra whose set of non-greatest elements has a greatest element ( and forms another Heyting algebra ) is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new greatest element.
46.
Every Heyting algebra whose set of non-greatest elements has a greatest element ( and forms another Heyting algebra ) is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new greatest element.
47.
Every Heyting algebra whose set of non-greatest elements has a greatest element ( and forms another Heyting algebra ) is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new greatest element.
48.
From this simple intuition it is already clear that domains often do not have a greatest element, since this would mean that there is an element that contains the information of " all " other elements-a rather uninteresting situation.
49.
Here, the join of an empty set of elements is defined to be the least element \ bigvee \ varnothing = 0, and the meet of the empty set is defined to be the greatest element \ bigwedge \ varnothing = 1.
50.
Every element " s " of a well-ordered set, except a possible greatest element, has a unique successor ( next element ), namely the least element of the subset of all elements greater than " s ".
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