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Algebraic Curves and Projective Geometry by Edoardo Ballico, Ciro Ciliberto

By Edoardo Ballico, Ciro Ciliberto

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We must prove that every poset-indexed diagram ) → ( − ???? = ???????? , ???????????? ∣ ???? ≤ ???? in ???? in Bool???? , with ???? a directed poset, has a colimit. The colimit ) → − → ( − (????, ???????? ∣ ???? ∈ ????) := lim ???? , where ???? = ???????? , ???????????? ∣ ???? ≤ ???? in ???? −→ → − in the category of Boolean algebras is characterized, among cocones above ???? , by the statements ∪ ????= (???????? “(???????? ) ∣ ???? ∈ ????) , ( ) ???????? (????) ≤ ???????? (????) ⇔ (∃???? ≥ ????) ???????????? (????) ≤ ???????????? (????) , for all ???? ∈ ???? and all ????, ???? ∈ ???????? 42 2 Boolean Algebras Scaled with Respect to a Poset (cf.

2 being closed under nonempty finite products. Hence we record for further use the following easy fact. 2. The category Bool???? has arbitrary nonempty finite products, and even arbitrary nonempty small products in case ???? is finite. Proof. Let ???? be a nonempty set and let (???????? ∣ ???? ∈ ????) be a family of ???? -scaled Boolean algebras. We set ???? := ∏ (???????? ∣ ???? ∈ ????) , ) ∏ ( (????) ???????? ∣ ???? ∈ ???? , ????(????) := )) ( ( ???? := ????, ????(????) ∣ ???? ∈ ???? . ) (????) ???????? ∣ ???? ∈ ???? , there are a finite subset ???????? of ???? and ) ∏ ( (????) ⋁ ???????? ∣ ???? ∈ ???????? such that 1???????? = (????????,???? ∣ ???? ∈ ???????? ).

3, ▽???? is finite in case ???? is finite. Every nonempty ▽-closed subset ???? in a pseudo join-semilattice ???? is also a pseudo join-semilattice, and ???? ▽???? ???? = ???? ▽???? ????, for all ????, ???? ∈ ????. 4. For a pseudo join-semilattice ???? , a positive integer ????, finite ∪ subsets ????0 , . . , ????????−1 of ???? , and ???? = ????

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