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In a totally ordered set, the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. This observation applies not only to totally ordered subsets of any partially ordered set, but also to their order theoretic generalization via ...
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
The real numbers, or in general any totally ordered set, ordered by the standard less-than-or-equal relation ≤, is a partial order. On the real numbers R {\displaystyle \mathbb {R} } , the usual less than relation < is a strict partial order.
This order is usually determined by the order in which the elements are added to the structure, but the elements can be rearranged in some contexts, such as sorting a list. For a structure that isn't ordered, on the other hand, no assumptions can be made about the ordering of the elements (although a physical implementation of these data types ...
Maxima and minima can also be defined for sets. In general, if an ordered set S has a greatest element m, then m is a maximal element of the set, also denoted as (). Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with (respect to order induced by T), then m is a least upper bound of S in T.
The least and greatest element of the whole partially ordered set play a special role and are also called bottom (⊥) and top (⊤), or zero (0) and unit (1), respectively. If both exist, the poset is called a bounded poset .
For example, for the array of values [−2, 1, −3, 4, −1, 2, 1, −5, 4], the contiguous subarray with the largest sum is [4, −1, 2, 1], with sum 6. Some properties of this problem are: If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array.
The NIST Dictionary of Algorithms and Data Structures [1] is a reference work maintained by the U.S. National Institute of Standards and Technology.It defines a large number of terms relating to algorithms and data structures.