A greatest element of a partially ordered subset must not be confused with " maximal elements " of the set, which are elements that are not smaller than any other elements.
42.
In the early 1960s minimalist artist Carl Andre described to Frampton the Dedekind cut, which partitions a totally ordered set into a set with no maximal element and a set with no minimal element.
43.
The "'demand correspondence "'maps any price p and any level of income m into the set of \ preceq-maximal elements of \ Gamma ( p, m ).
44.
The first approach assumes that preferences are at least " acyclic " ( which is necessary and sufficient for the preferences to have a maximal element on any " finite " subset ).
45.
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 ).
46.
"' Zorn's lemma "', also known as the "'Kuratowski Zorn lemma "', after mathematicians totally ordered subset ) necessarily contains at least one maximal element.
47.
Again, in infinite posets maximal elements do not always exist-the set of all " finite " subsets of a given infinite set, ordered by subset inclusion, provides one of many counterexamples.
48.
A semigroup " S " satisfies the "'maximal condition on congruences "'if any family of congruences on " S ", ordered by inclusion, has a maximal element.
49.
In his original proof, Mirsky constructs the same partition inductively, by choosing an antichain of the maximal elements of longest chains, and showing that the length of the longest chain among the remaining elements is reduced by one.
50.
This partially ordered set does not even have any maximal elements, since any " g " divides for instance 2 " g ", which is distinct from it, so " g " is not maximal.
