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Thus, another way of stating Sperner's theorem is that the width of the inclusion order on a power set is (⌊ / ⌋). A graded partially ordered set is said to have the Sperner property when one of its largest antichains is formed by a set of elements that all have the same rank. In this terminology, Sperner's theorem states that the partially ...
A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes ...
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 layers of the Earth, a differentiated planetary body. In planetary science, planetary differentiation is the process by which the chemical elements of a planetary body accumulate in different areas of that body, due to their physical or chemical behavior (e.g. density and chemical affinities).
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 ...
The James Webb Space Telescope and other observatories witnessed a massive explosion in space that created rare chemical elements, some of which are necessary for life.
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 ...
If that fails, saving an astronaut floating off into space might require several tethers hooked together, a SAFER, and, to be honest, a lot of luck. RELATED: Here's whats happening in space this year: