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An antichain in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable. A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest ...
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 ...
Three views of an antimatroid: an inclusion ordering on its family of feasible sets, a formal language, and the corresponding path poset. In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. [1]
In a partially ordered set there may be some elements that play a special role. The most basic example is given by the least element of a poset. For example, 1 is the least element of the positive integers and the empty set is the least set under the subset order. Formally, an element m is a least element if: m ≤ a, for all elements a of the ...
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
A locally finite poset is one in which every closed interval [a, b] = {x : a ≤ x ≤ b}is finite.. The members of the incidence algebra are the functions f assigning to each nonempty interval [a, b] a scalar f(a, b), which is taken from the ring of scalars, a commutative ring with unity.
800-290-4726 more ways to reach us. Sign in. Mail. 24/7 Help. ... England, said the report “paints a grim picture of human-caused disruption to the global water cycle, the most precious natural ...
Emoto claimed that water was a "blueprint for our reality" and that emotional "energies" and "vibrations" could change its physical structure. [14] His water crystal experiments consisted of exposing water in glasses to various words, pictures, or music, then freezing it and examining the ice crystals' aesthetic properties with microscopic photography. [9]