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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 ...
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 ...
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.)
A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set S is the set of its downwardly closed subsets ordered by inclusion. S is embedded in this (complete) lattice by mapping each element x to the lower set of elements that are less than or equal to x.
No way,” the priest continued, shaking his head and wagging his finger, eliciting more laughs from the pews. “So we’re all going to have to stand there in front of God.
Immanuel – "God with us," is a Biblical concept that deals with the concept of divine presence, often used by Christians as a title for Jesus; Incarnation (Christianity) – Believed of the second person of the Trinity, also known as God the Son or the Logos (Word), who "became flesh" by being conceived in the womb of Mary.
VATICAN CITY (Reuters) -Pope Francis said the story of Jesus' birth as a poor carpenter's son should instil hope that all people can make an impact on the world, as the pontiff on Tuesday led the ...