A maximal element is an element of a partially ordered set that has no larger neighbors. It’s also the only maximal element of that set. maximak
However, it’s important to note that a maximal element doesn’t necessarily have to be the only one. Moreover, it’s possible to have maximum cliques that aren’t the only ones.
Definition
In mathematics, a maximum is an element of a subset S displaystyle S of some preordered set that is greater than or equal to every other element of S. The minimum of S displaystyle S is defined dually as an element that is not greater than any other element of S.
In partially ordered sets, a maximal element is said to be the only one for which this definition applies. It is not necessary that a maximal element be the only maximum, however.
This distinction is important when considering a set that has some property of its own. For example, a clique of people who know each other is a maximal clique if adding anyone destroys its clique property.
Maximal and minimal can be used interchangeably in this sense, but the difference is subtle. A maximum element must be larger than (and hence comparable to) every other element of A,A, while a minimal element must be only larger than every other element of AA to which it is comparable.
Applications
The term maximal is often applied to sets that satisfy certain properties. For example, a clique that contains only people who know each other is a maximal clique. However, if you add people to that clique, it is no longer a maximum clique. This is a subtle distinction. Mathematicians make a similar distinction when discussing partially ordered sets. Basically, in this context, an element is considered to be maximal when there is no other element that is greater than it. This is a useful definition because it allows us to understand sets that don’t have to compare every pair of elements.
Terminology
In mathematics, a maximal element is an element that is the maximum of all elements of its kind. The term is sometimes used interchangeably with the word maximum, which refers to a set that is larger than any other set of its kind. It is often used in the context of partially ordered sets, where not all pairs of elements need to be comparable. For example, a clique is a maximum clique if adding anyone else to it destroys the “clique” property, meaning there are no other cliques that contain this group. Similarly, a sprint is a maximal sprint if it produces the most muscle building and is effective in burning fat.
