By Edoardo Ballico, Ciro Ciliberto

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**Example text**

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 ???? = ????