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By Arnon Avron, Anna Zamansky (auth.), Stefano Aguzzoli, Agata Ciabattoni, Brunella Gerla, Corrado Manara, Vincenzo Marra (eds.)

ISBN-10: 3540759387

ISBN-13: 9783540759386

Edited in collaboration with FoLLI, the organization of good judgment, Language and data, this e-book constitutes the 3rd quantity of the FoLLI LNAI subline. The 17 revised papers of this Festschrift quantity - released in honour of Daniele Mundici at the get together of his sixtieth birthday - comprise invited prolonged types of the main attention-grabbing contributions to the foreign convention at the Algebraic and Logical Foundations of Many-Valued Reasoning, held in Gargnano, Italy, in March 2006.

Daniele Mundici is commonly stated as a number one scientist in many-valued common sense and ordered algebraic constructions. within the final many years, his paintings has unveiled profound connections among good judgment and such assorted fields of study as sensible research, likelihood and degree conception, the geometry of toric types, piecewise linear geometry, and error-correcting codes. a number of well-known logicians, mathematicians, and desktop scientists attending the convention have contributed to this wide-ranging assortment with papers all variously with regards to Daniele's work.

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If M axA = M ax1 A = ∅, then, N = (NA (W1 ))⊥ . Proof. N ∩ NA (W1 ) ⊆ M∈MaxA M = {0}. Then N ⊆ (NA (W1 ))⊥ . On other hand if M ∈ M ax(A) \ M ax1 A, then NA (W1 ) M ; that implies (NA (W1 ))⊥ ⊆ M and so (N1 )⊥ ⊆ N . As an application of Proposition 41 we get a result which strengthens Theorem 3 of [4]. Indeed we have: Symmetric MV-Algebras 47 Proposition 48. Let A be an MV-algebra. Then the following statements are equivalent: 1. M axA = M ax1 A 2. ord(xy) = ∞ iff ord(x) = ∞ or ord(y) = ∞. 5 Topological Issues on Maximal Ideals of Type p In this section we provide a certain topological characterization of Uc (NA (Wp )).

Proposition 21. Let p ∈ P. , p−1 p , 1}; (ii) let a ∈]0, 1[. Wp (a) = 0 iff for exactly one k, 1 ≤ k ≤ p−1 2 , Wk,p (a) = 0; (iii) let A be an MV-algebra. If a ∈ M Vp (A), then Wp (a) = 0; if a ∈ RadA , then Wp (a) = a. Proof. (i) Let p = 2. W2 (a) = 0 if a ∈ {0, 1} or if a2 ∨ (a∗ )2 = 0. The latter implies that a2 = (a∗ )2 = 0, so a = a∗ and A ∼ = {0, 12 , 1}. If p > 2, the statement follows by Proposition 19. P. Belluce, A. Di Nola, and A. Lettieri (ii) It follows by Proposition 19. (iii) Let a ∈ M Vp (A).

Immediately we have: Lemma 16. Let F (z) be a nested monomial and A an M V -algebra. Then the following statements hold: (i) If a ∈ B(A), then F (a) = a. (ii) Let A be non Boolean. Then F (z) is the identity map on A iff m = n = 1 for all coefficients m and exponents n; (iii) If a ∈ RadA and for some exponent n, n > 1, then F (a) = 0. (iv) If If a ∈ (RadA)∗ and for some coefficient m, m > 1, then F (a) = 1. With the notations of (*) we get: Proposition 17. Let kp ∈ [0, 1], p ∈ P \ {2} and 0 < kp < 12 .

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Algebraic and Proof-theoretic Aspects of Non-classical Logics: Papers in Honor of Daniele Mundici on the Occasion of His 60th birthday by Arnon Avron, Anna Zamansky (auth.), Stefano Aguzzoli, Agata Ciabattoni, Brunella Gerla, Corrado Manara, Vincenzo Marra (eds.)


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