In mathematics, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of . The term least element is defined dually, that is, it is an element of that is smaller than every other element of
Let be a preordered set and let
An element is said to be if and if it also satisfies:
for all
By switching the side of the relation that is on in the above definition, the definition of a least element of is obtained. Explicitly, an element is said to be if and if it also satisfies:
for all
If is also a partially ordered set then can have at most one greatest element and it can have at most one least element. Whenever a greatest element of exists and is unique then this element is called greatest element of . The terminology least element of is defined similarly.
If has a greatest element (resp. a least element) then this element is also called (resp. ) of
Greatest elements are closely related to upper bounds.
Let be a preordered set and let
An is an element such that and for all Importantly, an upper bound of in is required to be an element of
If then is a greatest element of if and only if is an upper bound of in In particular, any greatest element of is also an upper bound of (in ) but an upper bound of in is a greatest element of if and only if it to
In the particular case where the definition of " is an upper bound of " becomes: is an element such that and for all which is to the definition of a greatest element given before.
Thus is a greatest element of if and only if is an upper bound of .
If is an upper bound of that is not an upper bound of (which can happen if and only if ) then can be a greatest element of (however, it may be possible that some other element a greatest element of ).
In particular, it is possible for to simultaneously have a greatest element for there to exist some upper bound of .
Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers.
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In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.
In mathematics, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of . The term least element is defined dually, that is, it is an element of that is smaller than every other element of Let be a preordered set and let An element is said to be if and if it also satisfies: for all By switching the side of the relation that is on in the above definition, the definition of a least element of is obtained.
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic, i.
Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a
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