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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 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 .
Then, by the well-ordering principle, there is a least element ; cannot be prime since a prime number itself is considered a length-one product of primes. By the definition of non-prime numbers, n {\displaystyle n} has factors a , b {\displaystyle a,b} , where a , b {\displaystyle a,b} are integers greater than one and less than n ...
An example is given by the above divisibility order |, where 1 is the least element since it divides all other numbers. In contrast, 0 is the number that is divided by all other numbers. Hence it is the greatest element of the order. Other frequent terms for the least and greatest elements is bottom and top or zero and unit. Least and greatest ...
The standard ordering ≤ of the natural numbers is a well ordering and has the additional property that every non-zero natural number has a unique predecessor. Another well ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers, and the usual ordering applies within the evens and the odds:
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.
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Other common names for the least element are bottom and zero (0). The dual notion, the empty lower bound, is the greatest element, top, or unit (1). Posets that have a bottom are sometimes called pointed, while posets with a top are called unital or topped. An order that has both a least and a greatest element is bounded.