Algebraically closed MV-algebras and their sheaf representation
An MV-algebra is an abelian monoid \(A= (A,1,\bullet)\) with an involution \(\neg\) satisfying the equations \(x\bullet 1= 1\) and \(\neg(\neg x\bullet y)\bullet y= \neg(\neg y\bullet x)\bullet x\). The standard MV-algebra is the unit real interval equipped with the operations \(\neg x= 1-x\) and \(x\bullet y= \min(0,x+ y- 1)\). The free \(n\)-generator MV-algebra consists of all continuous piecewise linear functions \(f:[0,1]^n\to [0,1]\) equipped with the pointwise operations of the standard MV-algebra. This is the McNaughton theorem. There exists a categorical equivalence \(\Gamma\) between unital abelian lattice-ordered groups (for short, unital \(\ell\)-groups) and MV-algebras. An MV-algebra \(A\) is divisible iff so is the unital \(\ell\)-group \((G,u)\) given by \(A= \Gamma(G,u)\). Using these basic facts, the authors prove that an algebraically closed MV-algebra is isomorphic to the MV-algebras of global sections in a sheaf over a Stone space, where the stalks are divisible totally ordered MV-algebras. A different characterization was provided by \textit{F. Lacava} [Boll. Unione Mat. Ital., VII. Ser., B 1, 703--712 (1987; Zbl 0647.06008); ibid. 11, No. 4, 961--972 (1997; Zbl 0886.06004)]. For background on MV-algebras see the monograph [\textit{R. L. O. Cignoli}, \textit{I. M. L. D'Ottaviano} and the reviewer, Algebraic foundations of many-valued reasoning. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)]. For the most recent developments see the present reviewer's book [Advanced Łukasiewicz calculus and MV-algebras. Berlin: Springer (2011; Zbl 1235.03002)].
algebraically closed sheaf representation
scientific article; zbMATH DE number 6137403