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  2. Order theory - Wikipedia

    en.wikipedia.org/wiki/Order_theory

    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 ...

  3. Well-order - Wikipedia

    en.wikipedia.org/wiki/Well-order

    Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element. The following binary relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds ...

  4. Well-ordering principle - Wikipedia

    en.wikipedia.org/wiki/Well-ordering_principle

    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 ...

  5. Greatest element and least element - Wikipedia

    en.wikipedia.org/wiki/Greatest_element_and_least...

    In the rational numbers, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound. In , the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.

  6. Maximal and minimal elements - Wikipedia

    en.wikipedia.org/wiki/Maximal_and_minimal_elements

    In abstract algebra, the concept of a maximal common divisor is needed to generalize greatest common divisors to number systems in which the common divisors of a set of elements may have more than one maximal element. In computational geometry, the maxima of a point set are maximal with respect to the partial order of coordinatewise domination.

  7. Ordinal number - Wikipedia

    en.wikipedia.org/wiki/Ordinal_number

    In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [1] A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used.

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  9. List of order structures in mathematics - Wikipedia

    en.wikipedia.org/wiki/List_of_order_structures...

    Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets ), orderings in which some pairs are comparable and others might not be

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