Algebraic foundations of many-valued reasoning
type: Item
instance of: scholarly article
title: Algebraic foundations of many-valued reasoning
zbMATH Open document ID: 0937.06009
author: Roberto Cignoli
author: Itala D'Ottaviano
author: Daniele Mundici
published in: Trends in Logic -- Studia Logica Library
publication date: 2000-01-04T00:00:00Z
review text: The book under review presents an outstanding monograph on MV-algebras and many-valued reasoning by worldwide known experts on this topic. MV-algebras entered mathematics in 1958 by C. C. Chang as algebraic analysis of many-valued logics. They give an algebraic framework for the infinite-valued propositional calculus of Łukasiewicz, and today MV-algebras are a well-developed theory with many nice and deep results and with many applications, e.g., in computer science. The accent is put on the algebraic point of view and on this basis the Łukasiewicz calculus is developed. The monograph consists of 10 chapters and a bibliography with 250 items. Chapter 1 is devoted to basic notions of MV-algebras. The original long list of Chang's axioms is reduced to six axioms: An MV-algebra is an algebra \((A; \oplus, \neg,0)\) of type \((2,1,0)\) such that, for all \(x,y, z \in A\), (MV1) \(x \oplus (y \oplus z) = (x \oplus y) \oplus z\); (MV2) \(x \oplus y = y \oplus x\); (MV3) \(x \oplus 0 = 0\); (MV4) \(\neg\neg x = x\); (MV5) \(x \oplus \neg 0= \neg 0\); (MV6) \(\neg(\neg x \oplus y) \oplus y = \neg(\neg y \oplus x) \oplus x.\) Putting \(x \leq y\) iff \(\neg x \oplus y = 1 :\neg 0\), we have that \((A,\leq)\) is a distributive lattice with the least and the greatest elements \(0\) and \(1\), respectively. For example, if \((G,u)\) is a unital abelian \(\ell\)-group with strong unit, then \(\Gamma(G,u) :=\{g \in G:0 \leq g \leq u\}\) equipped with \(g_1 \oplus g_2 := (g_1 + g_2) \wedge u\); \(\neg g:= u - g,\) gives a prototypical example of an MV-algebra. In particular, if \(G = \mathbb R\), \(u =1\), we obtain a very important case of MV-algebras. The fundamental result of Chang's Subdirect Representation Theorem is proved here using the theory of ideals. In addition, MV-equations are introduced, and it is proved that an MV-equation is satisfied by all MV-algebras iff it is satisfied by all MV-chains. This result is considerably strengthened in the second chapter. Chapter 2 …
reviewed by: Anatolij Dvurečenskij
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 06-02
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 06F20
zbMATH DE Number: 1382786
zbMATH Keywords: MV-algebra
zbMATH Keywords: lattice-ordered group
zbMATH Keywords: free MV-algebra
zbMATH Keywords: infinite-valued propositional calculus
zbMATH Keywords: McNaughton function
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1382786