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If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any prime number is a minimal element for it. In this poset, 60 is an upper bound (though not a least upper bound) of the subset {,,,}, which does not have any lower bound (since 1 is ...
Another way to classify incompatible elements is by mass (lanthanide series): light rare-earth elements (LREE) are La, Ce, Pr, Nd, and Sm, and heavy rare-earth elements (HREE) are Eu–Lu. Rocks or magmas that are rich, or only slightly depleted, in light rare-earth elements are referred to as "fertile", and those with strong depletions in LREE ...
The family of all subsets of an n-element set (its power set) can be partially ordered by set inclusion; in this partial order, two distinct elements are said to be incomparable when neither of them contains the other. The width of a partial order is the largest number of elements in an antichain, a set of pairwise incomparable elements ...
An antichain in is a subset of in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in . (However, some authors use the term "antichain" to mean strong antichain , a subset such that there is no element of the poset smaller than two distinct elements of the antichain.)
Now there are also elements of a poset that are special with respect to some subset of the order. This leads to the definition of upper bounds. Given a subset S of some poset P, an upper bound of S is an element b of P that is above all elements of S. Formally, this means that s ≤ b, for all s in S. Lower bounds again are defined by inverting ...
In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. If a directed set has a maximal element, it is also its greatest element, [proof 7] and hence its only maximal element. For a directed set without maximal or greatest elements, see examples 1 and 2 above.
The funniest movie ever made about teenage girls with crushes on Richard Nixon, the cult comedy Dick crashed and burned at the box office in 1999, but ended up fulfilling its rightful destiny: to ...
The human body and even its individual body fluids may be conceptually divided into various fluid compartments, which, although not literally anatomic compartments, do represent a real division in terms of how portions of the body's water, solutes, and suspended elements are segregated. The two main fluid compartments are the intracellular and ...