By Bacco M., Mocellin V.

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**Example text**

Q(q v) = (qq )v = vv = v. 9. (q q)(q q) = q (qq )q = q (vq) = q q. 7 (by (b), and 3) (I is an ideal) (I is minimal) (by 6) (by 7) (b) 10. (q v)q = q q = (q q)v = v. (b) 3 9,(b) (by 8, 10) 11. q v is a left and right inverse of q in I v. (d) 1. Let p ∈ I . 2. As before pI = I . (I is a minimal ideal) −1 3. K = {q ∈ I | pq = p} = Lp (p) is a nonempty closed subsemigroup of I . 9) 4. There exists an idempotent u ∈ J with pu = p. 9) 5. p ∈ I u. (by 4) Minimal sets and minimal right ideals 6. I = {I v | v ∈ J }.

2. There exists a surjective flow and semigroup homomorphism θ : E(X × X) → E(X). 10) 3. Let p, q ∈ βT with θ ( X×X (p)) = θ ( X×X (q)). 4. X (p) = X (q). 10) 5. Let (x, y) ∈ X × X. 6. 10 X×X (q)). 4 Flows and their enveloping semigroups 7. g2 ((x, y) X×X (p)) X×X (q)). 8. 10 X×X (p))) X×X (q))) X×X (q). 9. X×X (p) = X×X (q). 10. θ is an isomorphism. 12 Let: (i) (X, T ) be a flow, (ii) x0 ∈ X with x0 T = X (so that (X, T ) is point transitive), and (iii) f : E(X) → X be defined by f (p) = x0 p for all p ∈ E(X).

XT = M. (by 1, 2) (b) =⇒ (c) 1. Assume that xT = M for all x ∈ M and let U ⊂ X be open with U ∩ M = ∅. 2. xT ∩ U = ∅ for all x ∈ M. (by 1) 3. x ∈ U T for all x ∈ M. (by 2) (c) =⇒ (a) 1. Assume that (c) holds and N ⊂ M is a nonempty closed, invariant subset of M. 2. Let U = X \ N . 3. U is open. (by 1, 2) 4. N ∩ (U ∩ M)T ⊂ NT ∩ U = N ∩ U = ∅. (by 1, 2) 5. (U ∩ M)T = M. (by 1, 4) Minimal sets and minimal right ideals 6. U ∩ M = ∅. 7. N = M. 29 (by 1, 5) (by 1, 2, 6) We use the axiom of choice in the form of Zorn’s lemma both in the following proposition, to show that minimal sets exist, and later to show that certain semigroups contain idempotents.

### A bayesian justification for the linear pooling of opinions by Bacco M., Mocellin V.

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