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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 ...
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 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 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 ...
An element x of S embeds into the completion as its principal ideal, the set ↓ x of elements less than or equal to x. Then (↓ x) u is the set of elements greater than or equal to x, and ((↓ x) u) l = ↓ x, showing that ↓ x is indeed a member of the completion. The mapping from x to ↓ x is an order-embedding. [7]
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 ...
The series composition of P and Q, written P; Q, [7] P * Q, [2] or P ⧀ Q, [1] is the partially ordered set whose elements are the disjoint union of the elements of P and Q. In P; Q, two elements x and y that both belong to P or that both belong to Q have the same order relation that they do in P or Q respectively.
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 ...