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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 ...
The reason is that, in general, there are many different possible ways to draw a Hasse diagram for a given poset. The simple technique of just starting with the minimal elements of an order and then drawing greater elements incrementally often produces quite poor results: symmetries and internal structure of the order are easily lost.
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 parallel composition of P and Q, written P || Q, [7] P + Q, [2] or P ⊕ Q, [1] is defined similarly, from the disjoint union of the elements in P and the elements in Q, with pairs of elements that both belong to P or both to Q having the same order as they do in P or Q respectively. In P || Q, a pair x, y is incomparable whenever x belongs ...
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 ...
Let n be a positive integer, and let P be the partial order on the elements a i and b i (for 1 ≤ i ≤ n) in which a i ≤ b j whenever i ≠ j, but no other pairs are comparable. In particular, a i and b i are incomparable in P; P can be viewed as an oriented form of a crown graph. The illustration shows an ordering of this type for n = 4.
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 ...