##
**Lukasiewicz-Moisil algebras.**
*(English)*
Zbl 0726.06007

Annals of Discrete Mathematics, 49. Amsterdam etc.: North-Holland. xv, 583 p. Dfl. 250.00 (1991).

Many-valued logic was introduced by J. Łukasiewicz, who defined a three-valued propositional calculus in 1920. Later, the same author considered n-valued propositional calculi, and, independently, E. Post studied a different n-valued propositional calculus.

Three-valued and four-valued Łukasiewicz algebras were inroduced by G. Moisil with the purpose of obtaining the associated Lindenbaum-Tarski algebras of the corresponding logics of Łukasiewicz, and later he introduced the notion of \(n\)-valued Łukasiewicz algebras. But whereas the three-valued and four-valued Łukasiewicz algebras are the Lindenbaum-Tarski algebras of the corresponding logics, A. Rose observed that this is not true for \(n\geq 5\) because the Łukasiewicz implication cannot be defined in terms of disjunction, conjunction, negation and endomorphisms in n-valued Łukasiewicz algebras, \(n\geq 5\). For this reason, R. Cignoli called these algebras Moisil algebras. On the other hand, Moisil introduced a propositional calculus whose corresponding Lindenbaum-Tarski algebra is precisely an n-valued Łukasiewicz algebra. Then, while Łukasiewicz algebras originated in Łukasiewicz logics, they were created and developed by Moisil. That is why the authors name these algebras Łukasiewicz-Moisil algebras (LM-algebras).

This monograph can be considered as a textbook on the algebraic side of the theory of LM-algebras, and also covers some applications to switching theory. The authors provide a detailed presentation of some of the most important papers on the subject and summarize others. In the first two chapters they present all the background needed for the understanding of the monograph: lattices, universal algebra, categories and topological dualities in lattice theory. After providing a thorough introduction to LM-algebras, they investigate Post algebras, axled LM-algebras, LM- algebras and Heyting algebras. Several representations of LM-algebras are presented. Also, monadic and polyadic LM-algebras are investigated. A chapter deals with Łukasiewicz logics: the three-valued Łukasiewicz logic in the Wajsberg axiomatization, the \(n\)-valued Łukasiewicz logics in the Cignoli axiomatization, and a logic whose theorems are the propositions true for all \(i\in I\) greater than a fixed \(k\in I.\)

The detailed presentation will make this monograph useful for a semester course, but it will also be valuable as a reference for mathematicians, logicians and computer scientists. In addition, there is stimulating material in this monograph for further investigation.

Three-valued and four-valued Łukasiewicz algebras were inroduced by G. Moisil with the purpose of obtaining the associated Lindenbaum-Tarski algebras of the corresponding logics of Łukasiewicz, and later he introduced the notion of \(n\)-valued Łukasiewicz algebras. But whereas the three-valued and four-valued Łukasiewicz algebras are the Lindenbaum-Tarski algebras of the corresponding logics, A. Rose observed that this is not true for \(n\geq 5\) because the Łukasiewicz implication cannot be defined in terms of disjunction, conjunction, negation and endomorphisms in n-valued Łukasiewicz algebras, \(n\geq 5\). For this reason, R. Cignoli called these algebras Moisil algebras. On the other hand, Moisil introduced a propositional calculus whose corresponding Lindenbaum-Tarski algebra is precisely an n-valued Łukasiewicz algebra. Then, while Łukasiewicz algebras originated in Łukasiewicz logics, they were created and developed by Moisil. That is why the authors name these algebras Łukasiewicz-Moisil algebras (LM-algebras).

This monograph can be considered as a textbook on the algebraic side of the theory of LM-algebras, and also covers some applications to switching theory. The authors provide a detailed presentation of some of the most important papers on the subject and summarize others. In the first two chapters they present all the background needed for the understanding of the monograph: lattices, universal algebra, categories and topological dualities in lattice theory. After providing a thorough introduction to LM-algebras, they investigate Post algebras, axled LM-algebras, LM- algebras and Heyting algebras. Several representations of LM-algebras are presented. Also, monadic and polyadic LM-algebras are investigated. A chapter deals with Łukasiewicz logics: the three-valued Łukasiewicz logic in the Wajsberg axiomatization, the \(n\)-valued Łukasiewicz logics in the Cignoli axiomatization, and a logic whose theorems are the propositions true for all \(i\in I\) greater than a fixed \(k\in I.\)

The detailed presentation will make this monograph useful for a semester course, but it will also be valuable as a reference for mathematicians, logicians and computer scientists. In addition, there is stimulating material in this monograph for further investigation.

Reviewer: M.Abad (Bahia Blanca)

### MSC:

06D30 | De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) |

03G20 | Logical aspects of Łukasiewicz and Post algebras |

03B50 | Many-valued logic |

94C10 | Switching theory, application of Boolean algebra; Boolean functions (MSC2010) |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

06D05 | Structure and representation theory of distributive lattices |

06D20 | Heyting algebras (lattice-theoretic aspects) |

06D25 | Post algebras (lattice-theoretic aspects) |

03G25 | Other algebras related to logic |

08B30 | Injectives, projectives |

06-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures |