##
**Algebraic foundations of many-valued reasoning.**
*(English)*
Zbl 0937.06009

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 is devoted to Chang’s Completeness Theorem. The above mapping \(\Gamma\) from the category \(\mathcal A \) whose objects are the unital abelian \(\ell\)-groups into the category \({\mathcal {MV}}\) of MV-algebras gives a functor which plays a fundamental role, because it defines even a natural equivalence as is proved in a subsequent chapter. For this reason, good sequences will be of special interest. We say that a sequence a \(= (a_1,a_2,\ldots)\) of elements of an MV-algebra is good iff, for each \(i \geq 1\), \(a_i \oplus a_{i+1} = a_i\) and there is an integer \(n\) such that \(a_r =0\) for all \(r \geq n.\) Good sequences are converted into a partially ordered monoid \(M_A\), and this into a Chang unital \(\ell\)-group \((G_A,u_A)\) with strong unit \(u_A\); in particular, any MV-algebra \(A\) is isomorphic with \(\Gamma(G_A,u_A)\). Consequently, Chang’s Completeness Theorem saying that an equation holds in \([0,1]\) iff it holds in every MV-algebra, is proved. Chapter 3 is dedicated to free MV-algebras. A special role is played by the free MV-algebras \(\text{Free}_n\) over \(n\) generators. \(\text{Free}_n\) can be easily described by piecewise linear continuous functions over \([0,1]^n\) with values in the interval \([0,1]\). Such functions are so-called McNaughton functions, and they are studied in detail. In many cases, very important MV-algebras (simple and semisimple) can be described as MV-subalgebras of continuous fuzzy sets over some compact Hausdorff spaces.

The radical, \(\text{Rad}(A)\), is defined as the intersection of all maximal ideals of \(A\), it gives important information on \(A\). It consists of all infinitesimals, i.e., of all elements \(a\) such that \(na \leq \neg a\) for each \(n \geq 1.\) Łukasiewicz \(\infty\)-valued calculus is described in Chapter 4. In the early twenties Łukasiewicz introduced a system of logic in which propositions admit as truth values real numbers between 0 and 1. The main connectives are \(\text{implication }\to\) and \(\text{negation }\neg\), such that \(x \to y := \min(1,1-x+y)\) and \(\neg x = 1- x.\) These connectives can be rewritten in terms of MV-algebras as \(x \to y = \neg x \oplus y.\) Using results of Chapter 3, it is proved that all tautologies are obtainable from a certain set of initial tautologies (corresponding to the MV-axioms) by a finite number of applications of modens ponens, and an effective procedure to decide whether a proposition is a tautology. For this reason, Wajsberg algebras and Lindenbaum algebras are studied. In particular, it is shown that every countable MV-algebra is the Lindenbaum algebra of some theory. Ulam’s game is presented in Chapter 5. This game goes back to Ulam. The authors consider the variant of Twenty Questions where \(n-2\) lies or errors are allowed. In the Ulam game with \(m\) lies, our knowledge is presented by the function \(\sigma : S \to \{0,1,\ldots,m-1\}\), where \(S\) is a finite set of numbers. Such problems are closely connected with finding an optimal strategy in the Ulam game with \(m\) lies or finding an optimal \(m\)-error correcting code. MV-algebras can be allowed as algebras of states of knowledge in generalized Ulam games, where the number of lies may depend on the secret number \(x \in S\), and \(S\) itself may be infinite. Truth values may be irrational numbers in \([0,1]\), or even nonstandard real numbers. Chapter 6 is a continuation of lattice-theoretical properties of MV-algebras. Here minimal prime ideals are studied together with Stonean ideals, archimedean elements, and hyperarchimedean algebras. In addition, complete MV-algebras and complete distributivity are studied. In Chapter 7, the crucial fact that the category of MV-algebras and the category of abelian unital \(\ell\)-groups are categorically equivalent via the functor \(\Gamma\) defined above is established (this is the famous Mundici Representation Theorem). As a corollary, a genuine addition can be uniquely recovered from the MV-algebraic structure. Perfect MV-algebras are MV-algebras \(A\) such that each element \(x \in A\) belongs either to \(\text{Rad}(A)\) or to \(\neg \text{Rad}(A) :=\{\neg a:\;a \in \text{Rad}(A)\}\). For such MV-algebras it is shown that the category of perfect perfect MV-algebras is categorically equivalent with the category of all \(\ell\)-groups. Chapter 8 is dedicated to the description of all varieties of MV-algebras. Komori’s classification and varieties generated by finite chains are presented. Advanced topics are given in Chapter 9. The first part deals with disjunctive minimal forms in the infinite-valued calculus of Łukasiewicz. The relationship between MV-algebras and approximately finite-dimensional \(C^*\)-algebras is presented. Finally, the important Di Nola Representation Theorem is given, which says that every MV-algebra \(A\) is an algebra of \([0,1]^*\)-valued functions over some set, where \([0,1]^*\) is an ultrapower of \([0,1]\), depending only on the cardinality of \(A\). The last chapter is Further Reading, where the authors outline further ways of studying MV-algebras, like states, observables, product, probability, etc. All chapters contain bibliographical remarks.

The wonderful book is addressed to computer scientists, mathematicians, logicians wishing to get acquainted with a compact body of beautiful theory, results and methodologies on MV-algebras, that have found applications in the handling of uncertain information, connecting many areas of mathematics like lattice-ordered groups, \(C^*\)-algebras, lattices, algebra, geometry of numbers, model theory, polyhedra, etc. It is a welcome addition to the literature.

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 is devoted to Chang’s Completeness Theorem. The above mapping \(\Gamma\) from the category \(\mathcal A \) whose objects are the unital abelian \(\ell\)-groups into the category \({\mathcal {MV}}\) of MV-algebras gives a functor which plays a fundamental role, because it defines even a natural equivalence as is proved in a subsequent chapter. For this reason, good sequences will be of special interest. We say that a sequence a \(= (a_1,a_2,\ldots)\) of elements of an MV-algebra is good iff, for each \(i \geq 1\), \(a_i \oplus a_{i+1} = a_i\) and there is an integer \(n\) such that \(a_r =0\) for all \(r \geq n.\) Good sequences are converted into a partially ordered monoid \(M_A\), and this into a Chang unital \(\ell\)-group \((G_A,u_A)\) with strong unit \(u_A\); in particular, any MV-algebra \(A\) is isomorphic with \(\Gamma(G_A,u_A)\). Consequently, Chang’s Completeness Theorem saying that an equation holds in \([0,1]\) iff it holds in every MV-algebra, is proved. Chapter 3 is dedicated to free MV-algebras. A special role is played by the free MV-algebras \(\text{Free}_n\) over \(n\) generators. \(\text{Free}_n\) can be easily described by piecewise linear continuous functions over \([0,1]^n\) with values in the interval \([0,1]\). Such functions are so-called McNaughton functions, and they are studied in detail. In many cases, very important MV-algebras (simple and semisimple) can be described as MV-subalgebras of continuous fuzzy sets over some compact Hausdorff spaces.

The radical, \(\text{Rad}(A)\), is defined as the intersection of all maximal ideals of \(A\), it gives important information on \(A\). It consists of all infinitesimals, i.e., of all elements \(a\) such that \(na \leq \neg a\) for each \(n \geq 1.\) Łukasiewicz \(\infty\)-valued calculus is described in Chapter 4. In the early twenties Łukasiewicz introduced a system of logic in which propositions admit as truth values real numbers between 0 and 1. The main connectives are \(\text{implication }\to\) and \(\text{negation }\neg\), such that \(x \to y := \min(1,1-x+y)\) and \(\neg x = 1- x.\) These connectives can be rewritten in terms of MV-algebras as \(x \to y = \neg x \oplus y.\) Using results of Chapter 3, it is proved that all tautologies are obtainable from a certain set of initial tautologies (corresponding to the MV-axioms) by a finite number of applications of modens ponens, and an effective procedure to decide whether a proposition is a tautology. For this reason, Wajsberg algebras and Lindenbaum algebras are studied. In particular, it is shown that every countable MV-algebra is the Lindenbaum algebra of some theory. Ulam’s game is presented in Chapter 5. This game goes back to Ulam. The authors consider the variant of Twenty Questions where \(n-2\) lies or errors are allowed. In the Ulam game with \(m\) lies, our knowledge is presented by the function \(\sigma : S \to \{0,1,\ldots,m-1\}\), where \(S\) is a finite set of numbers. Such problems are closely connected with finding an optimal strategy in the Ulam game with \(m\) lies or finding an optimal \(m\)-error correcting code. MV-algebras can be allowed as algebras of states of knowledge in generalized Ulam games, where the number of lies may depend on the secret number \(x \in S\), and \(S\) itself may be infinite. Truth values may be irrational numbers in \([0,1]\), or even nonstandard real numbers. Chapter 6 is a continuation of lattice-theoretical properties of MV-algebras. Here minimal prime ideals are studied together with Stonean ideals, archimedean elements, and hyperarchimedean algebras. In addition, complete MV-algebras and complete distributivity are studied. In Chapter 7, the crucial fact that the category of MV-algebras and the category of abelian unital \(\ell\)-groups are categorically equivalent via the functor \(\Gamma\) defined above is established (this is the famous Mundici Representation Theorem). As a corollary, a genuine addition can be uniquely recovered from the MV-algebraic structure. Perfect MV-algebras are MV-algebras \(A\) such that each element \(x \in A\) belongs either to \(\text{Rad}(A)\) or to \(\neg \text{Rad}(A) :=\{\neg a:\;a \in \text{Rad}(A)\}\). For such MV-algebras it is shown that the category of perfect perfect MV-algebras is categorically equivalent with the category of all \(\ell\)-groups. Chapter 8 is dedicated to the description of all varieties of MV-algebras. Komori’s classification and varieties generated by finite chains are presented. Advanced topics are given in Chapter 9. The first part deals with disjunctive minimal forms in the infinite-valued calculus of Łukasiewicz. The relationship between MV-algebras and approximately finite-dimensional \(C^*\)-algebras is presented. Finally, the important Di Nola Representation Theorem is given, which says that every MV-algebra \(A\) is an algebra of \([0,1]^*\)-valued functions over some set, where \([0,1]^*\) is an ultrapower of \([0,1]\), depending only on the cardinality of \(A\). The last chapter is Further Reading, where the authors outline further ways of studying MV-algebras, like states, observables, product, probability, etc. All chapters contain bibliographical remarks.

The wonderful book is addressed to computer scientists, mathematicians, logicians wishing to get acquainted with a compact body of beautiful theory, results and methodologies on MV-algebras, that have found applications in the handling of uncertain information, connecting many areas of mathematics like lattice-ordered groups, \(C^*\)-algebras, lattices, algebra, geometry of numbers, model theory, polyhedra, etc. It is a welcome addition to the literature.

Reviewer: A.Dvurečenskij (Bratislava)

### MSC:

06D35 | MV-algebras |

03B50 | Many-valued logic |

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03G25 | Other algebras related to logic |

06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |