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

**`**Meanwhile, a wealthy constitution conception for MV-algebras has been constructed, bearing on them e.g. to *l*-groups and to nonstandard reals. the current publication develops those issues 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and provides a coherent presentation of the middle result of the final 15 years or so, additionally including unpublished fabric of the authors.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 should be an necessary source.**'****Mathematical reports, 2001**

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O. , Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000 32 CHAPTER 2. 3) t+(xAy)=(t+x)A(t+y). In every o-group xVy = max{x,y} and xAy = min{x,y}; o-groups are particular cases of f-groups. 6) lxi =d,eJ x+ + x- = x V -x. , an element 0 ~ u E G such that for each x E G there is an integer n ~ 0 with lxi ~ nu. 1 Let G be an f-group. For any element u E G, u > 0, (not necessarily u being a strong unit of G) we let [0, u] =d,eJ {x E G I 0 ~ x ~ u}, and for each x, y E [0, u], x E9 y =d,eJ u A (x + y), and -,x =d,eJ u - x.

Fs(i)) a Ai (fl (i), ... , fs(i)) = aA(fI"'" fs)(i) , = whence TA(JI, . , fs) = aA(JI, . ,fs). 6 Let A be the subdirect product of a family {AihEl of MV-algebras; let r = a be an MV-equation. Then A F T = a if and only if Ai F T = a for each i EI. CHAPTER 1. 1. Suppose that A F r = (J'. Since for each i E I, 7ri 0 h maps A onto Ai, it follows that Ai Fr = (J. Conversely, suppose that Ai F r = (J' for all i EI. By the above lemma, [liEf Ai F r = (J', and since h(A) is a sub algebra of [liEf Ai, h(A) F r = (J'.

The basis is trivial. For the induction step, let el, ... ,em+1 in RT have the property that for any real numbers 0 ~ Ab ... , Am +l, ifE~tlAiei = 0, then Al = ... = Am +l = O. A fortiori, for any 0 ~ 111, ... , 11m with E~l l1iei = 0, we must have 111 = ... = 11m = O. Then by induction hypothesis there is a vector u E RT such that ei· u > 0 for each i = 1, ... , m. We now argue by cases: Gase 1: e m+1· u > O. Then the desired conclusion follows upon letting v=u. e m+l· u = O. Then one gets the desired result by letting V = au + em+1, where a = 1 + max{ei~:~~t li = 1, ...