By Edoardo Ballico, Ciro Ciliberto

ISBN-10: 3540515097

ISBN-13: 9783540515098

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**Extra resources for Algebraic Curves and Projective Geometry**

**Example text**

7 The product poset (P1 × P2 , ≤) is the poset whose elements are the elements of the Cartesian product P1 × P2 , where element (a1 , a2 ) is declared to be less-than-or-equal to (b1 , b2 ) if and only if a1 ≤ b1 and also a2 ≤ b2 . It should be clear that this notion can be extended to any collection of posets {(Pσ , ≤)|σ ∈ I } to form a direct product of posets.. Its elements are the elements of the Cartesian product σ∈I Pσ —that is, the functions f : I → U , where U is the disjoint union of the sets Pσ with the property that at any σ in I, f always assumes a 6 In a great deal of the literature, sets of pairwise incomparable elements are called independent.

11 4. The partition set: n . Suppose X is a set of just n elements. Recall that a partition of X is a collection π := {Y1 , . . , Yk } of non-empty subsets Y j whose join is X but which pairwise intersect at the empty set. The subsets Y j are called the components of the partition π. Suppose π1 := {Yi |i ∈ I } is a partition of X and π = {Z k |k ∈ K } is a second partition. We say that partition π refines partition π if and only if there exists a partition I = J1 + · · · Jk of the index set, such that Yi := Z .

Since both x and y lie in the totally ordered set Wμ , we write x ≤ y or y ≤ x according as x ≤μ y or y ≤μ x. In other words, in comparing two elements of WC , we utilize the comparison that works in any of the posets (Wλ , ≤λ ) or (Wμ , ≤μ ) that may contain both of them. The comparisons will always be consistent since each poset is an initial segment of any poset above it in the chain. Next, we must show that the poset (WC , ≤) is well-ordered. For that purpose, consider a non-empty subset S of WC .

### Algebraic Curves and Projective Geometry by Edoardo Ballico, Ciro Ciliberto

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