Algebra Qualifying Exam University of Miami
نویسنده
چکیده
(d) We say that an element p ∈ P (resp. q ∈ Q) is ∗∗-closed if p∗∗ = p (resp. q∗∗ = q). Prove that the Galois connection ∗ : P Q : ∗ restricts to an order-reversing bijection between ∗∗-closed elements. Proof. Let P ∗ ⊆ P and Q∗ ⊆ Q denote the images of the functions ∗ : Q → P and ∗ : P → Q, respectively. I claim that P ∗ ⊆ P is precisely the subset of ∗∗-closed elements. Indeed, if p∗∗ = p then p = (p∗)∗ is the image of p∗. Conversely, if p = q∗ for some q ∈ Q then by part (c) we have p∗∗ = (q∗)∗∗ = (q∗) = p. Similarly, we can show that Q∗ ⊆ Q is the subset of ∗∗-closed elements in Q. It follows immediately that the functions ∗ : Q∗ P ∗ : ∗ are inverse to each other, hence they are bijections. [The fact that they reverse order follows from (b).]
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