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All pairs of elements from are comparable and every element of is a greatest element (and thus also a maximal element) of (,). So in particular, if R {\displaystyle R} has at least two elements then ( R , ≤ ) {\displaystyle (R,\leq )} has multiple distinct greatest elements.
A minimal element of a subset of some preordered set is defined dually as an element of that is not greater than any other element in . The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.
Likewise, a greatest element of a partially ordered set (poset) is an upper bound of the set which is contained within the set, whereas the maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A), then m = b. Any least element or greatest element of a poset is unique, but a poset can have several minimal or ...
The supremum (abbreviated sup; pl.: suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to each element of , if such an element exists. [1] If the supremum of S {\displaystyle S} exists, it is unique, and if b is an upper bound of S {\displaystyle S} , then the supremum of S {\displaystyle S} is ...
Similarly, an element is a minimal element if there is no element such that <. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements.
Greatest element (maximum, top, unit), Least element (minimum, bottom, zero) Maximal element, minimal element; Upper bound. Least upper bound (supremum, join) Greatest lower bound (infimum, meet) Limit superior and limit inferior; Irreducible element; Prime element; Compact element
As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. golden spiral In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. [52]
In statistics, the sample maximum and sample minimum, also called the largest observation and smallest observation, are the values of the greatest and least elements of a sample. [1] They are basic summary statistics , used in descriptive statistics such as the five-number summary and Bowley's seven-figure summary and the associated box plot .