## Algebraic Foundations of Many-Valued Reasoning by Roberto L. O. Cignoli, Itala M. L. D’Ottaviano, Daniele

By Roberto L. O. Cignoli, Itala M. L. D’Ottaviano, Daniele Mundici

**`**Meanwhile, a wealthy constitution conception for MV-algebras has been constructed, concerning them e.g. to *l*-groups and to nonstandard reals. the current ebook develops those concerns intimately, and provides a coherent presentation of the middle result of the final 15 years or so, additionally including unpublished fabric of the authors. For destiny paintings on MV-algebras, this monograph may be an imperative source.**'****Mathematical experiences, 2001**

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6. 6 27 MV-chains In this section we collect several results on totally ordered MV-algebras, to be used in the next chapter. y; = (vii) 1/ x EB y = 1 and x EB z < 1 then (x 0 y) EB z = (x EB z) 0 y. x. xEB(z0x) = max(-,x,z). x, z), whence y = z. Condition (iii) is an immediate consequence of (i) and (ii). 4(i). Condition (v) follows from (iii). Prom (v) one immediately obtains (vi). y V (x EB z) = x EB z, whence (vii) follows from (iii). 3 it follows that every MV-algebra satisfies conditions (ii) and (vi).

B, a e b). Gase 1: b ~ a. b = 1, c = (lp-q,a e b) and c + b = (1 P , (a e b) EB b) = (1 P , b V a, 0) = (1 P , a) = a. b, (a e b) 0 Gase 2: b> a. \ b) = (F, a) = a. 2. 3. 4 Let a and b be good sequences. (i) If b $ athen there is a unique good sequence c such that b+c = a. 15) (-,a) = (1) - (a). (iii) The order is translation invariant, in the sense that b $ a implies b + d $ a + d for every good sequence d. 1 (i). 5 Let a = (al, . , an, ... ) and b = (bI, ... ,bn , ... ) be good sequences of an MV-algebra A.

Ii) Analogously, the good sequence is the infimum of a and b. 16) ((a) + (b)) /\ (1) = (a $ b). 11). 4 CHAPTER 2. 5, one can routinely obtain an f-group GA such that M A is isomorphie, both as a monoid and as a lattice, to the positive cone GA +. To this purpose, mimicking the construction of Z from N, let us agree to say that a pair of good sequences (a, b) is equivalent to another pair (a', b') iff a + b' = a' + b. 1(i). Notation: The equivalence class of the pair (a, b) shall be denoted by [a, b].