Monadic MV-algebras are equivalent to monadic \(\ell \)-groups with strong unit
Monadic MV-algebras were introduced by J. D. Rutledge as an algebraic model for the (monadic) predicate calculus of Łukasiewicz infinite-valued logic with a single individual variable. These algebras have been studied by many authors, such as Di Nola, Grigolia, Belluce, Lettieri. It is well known, by Mundici's famous theorem, that the MV-algebras are categorically equivalent to the \(\ell\)-groups with strong unit. More precisely, if \(G\) is an abelian \(\ell\)-group and \(u\) is a strong unit of \(G\), then the interval \([0, u]\) can be endowed with the structure of an MV-algebra, denoted \((\Gamma, u)\). The functor \(\Gamma\), defined in this way, establishes the above categorical equivalence. The aim of the present paper is to extend Mundici's functor \(\Gamma\) to the category of monadic MV-algebras. In order to do that, the authors define quantifiers on the lattice monoid of good sequences \({M_A}\) of an MV-algebra \(A\), introduce the notion of monadic \(\ell\)-group, and construct the monadic Chang \(\ell\)-group. Namely, if \({M_{\exists A}}\) denotes the lattice monoid of good sequences of elements of \(\exists A\), then define: \(\exists_M, \forall_M:M_A \longrightarrow M_A\) by \(\exists_M(M_A)=M_{\exists A}\) and \(\forall_M a=\neg \exists_M \neg a\) for all \(a\in M_A\). Quantifiers on Chang's \(\ell\)-group \(G_A\) of a monadic MV-algebra \(A\) are defined as follows: \(\exists_G, \forall_G:{G_A} \rightarrow {G_A}\) by \(\exists_G([\bar a, \bar b])=[\exists_M((\bar a\vee \bar b)-\bar b), \forall_M((\bar a \vee \bar b)-\bar a)]\) and \(\forall_G([\bar a, \bar b])=\neg \exists_G \neg [\bar a, \bar b]\). If \((G; +, -, 0, \leq)\) is an \(\ell\)-group and \(u>0\) is a fixed element of \(G\), then the authors define the notion of monadic \(\ell\)-group \({G}=(G; +, -, 0, \leq, u, \exists)\). For a monadic MV-algebra \({A}\), the structure \((G_A; [(1), (0)], \exists_G)\) is a monadic \(\ell\)-group, which is called the monadic Chang \(\ell\)-group. The natural equivalence between the category of monadic MV-algebras and the category of monadic \(\ell\)-groups with strong unit is established by the functor \({\Gamma_{\exists}}\). If \(f:(G;u, \exists)\longrightarrow (H;v, \exists {'})\) is a monadic \(\ell\)-group homomorphism and \(\Gamma_{\exists}(f)=f_{\mid [0, u]}\), then \(\Gamma_{\exists}\) is a functor from the category of monadic \(\ell\)-groups into the category of monadic MV-algebras. Finally, some applications of this equivalence are given. More precisely, the authors construct an order isomorphism between the set of monadic ideals of the monadic MV-algebra \({A}\) and the ordered set of monadic \(\ell\)-ideals of a monadic \(\ell\)-group \({G}\). I think that the article contains valuable results and is a starting point for other studies on this subject.
scientific article; zbMATH DE number 6013466