Hyperfinite MV-algebras.

*(English)*Zbl 1276.06006The paper concerns hyperfinite MV-algebras, which are infinite models of the theory of finite MV-algebras. MV-algebras were introduced in the fifties by C. C. Chang in [Trans. Am. Math. Soc. 88, 467–490 (1958; Zbl 0084.00704)] as the algebraic counterpart of Łukasiewicz infinite-valued logic. The article is divided into eight sections, and as the authors explain in the first one, there are three main results.

As the first main result, it is shown that every hyperfinite MV-algebra is elementarily equivalent to a (finite respectively infinite) direct product of finite respectively hyperfinite MV-chains (Sections 2 and 3).

As the second main result, it is shown that the theory of all finite MV-algebras is recursively and co-recursively enumerable (Sections 4 and 5). Also, recursive axiomatizations of finite respectively hyperfinite MV-algebras are presented. As a consequence, it holds that the theory of all finite MV-algebras is decidable. This last result is particularly interesting, because the general case of the theory of all MV-algebras is undecidable (see [H. W. Buff, Algebra Univers. 21, 234–249 (1985; Zbl 0563.03039)]).

As the third main result, it is shown that if \(A\) is a hyperfinite MV-algebra, then \(A/\mathrm{Rad}(A)\) is infinite (Section 6).

Finally, in Section 7 and Section 8 many other results concerning hyperfinite MV-algebras are shown, and some open problems are discussed.

As the first main result, it is shown that every hyperfinite MV-algebra is elementarily equivalent to a (finite respectively infinite) direct product of finite respectively hyperfinite MV-chains (Sections 2 and 3).

As the second main result, it is shown that the theory of all finite MV-algebras is recursively and co-recursively enumerable (Sections 4 and 5). Also, recursive axiomatizations of finite respectively hyperfinite MV-algebras are presented. As a consequence, it holds that the theory of all finite MV-algebras is decidable. This last result is particularly interesting, because the general case of the theory of all MV-algebras is undecidable (see [H. W. Buff, Algebra Univers. 21, 234–249 (1985; Zbl 0563.03039)]).

As the third main result, it is shown that if \(A\) is a hyperfinite MV-algebra, then \(A/\mathrm{Rad}(A)\) is infinite (Section 6).

Finally, in Section 7 and Section 8 many other results concerning hyperfinite MV-algebras are shown, and some open problems are discussed.

Reviewer: Matteo Bianchi (Milano)

##### MSC:

06D35 | MV-algebras |

03B50 | Many-valued logic |

03B25 | Decidability of theories and sets of sentences |

03C10 | Quantifier elimination, model completeness, and related topics |

03C60 | Model-theoretic algebra |