Volume 65, Issue 4

In this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.

In this short paper we will discuss on saturated and κ-saturated models of many-valued (t-norm based fuzzy) logics. Using these peculiar structures we show a representation theorem à la Di Nola for several classes of algebras including MV, Gödel, product, BL, NM and WNM-algebras. Then, still using (κ)-saturated algebras, we finally show that some relevant subclasses of algebras related to many-valued logics also enjoy the joint embedding property and the amalgamation property.

We study ℍ-perfect pseudo MV-algebras, that is, algebras which can be split into a system of ordered slices indexed by the elements of an subgroup ℍ of the group of the real numbers. We show when they can be represented as a lexicographic product of ℍ with some ℓ-group. In addition, we show also a categorical equivalence of this category with the category of ℓ-groups.

Riesz MV-algebras are a variety of algebras strongly connected to Riesz spaces. In this short article we investigate some elimination properties of the first-order theory RMV of linearly ordered Riesz MV-algebras and show that RMV admits elimination of quantifiers and uniform elimination of imaginary elements. In the process, we also prove several other results such as modelcompleteness, o-minimality, definability of Skolem functions, and a version of the Di Nola Representation Theorem for Riesz MV-algebras.

MV-algebras and Riesz MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit and, respectively, with Riesz spaces (vector-lattices) with strong unit. A standard construction in the literature of lattice-ordered groups is the vector-lattice hull of an archimedean latticeordered group. Following a similar approach, in this paper we define the Riesz hull of a semisimple MV-algebra.

The amalgamation property (AP) is of particular interest in the study of residuated lattices due to its relationship with various syntactic interpolation properties of substructural logics. There are no examples to date of non-commutative varieties of residuated lattices that satisfy the AP. The variety SemRL of semilinear residuated lattices is a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the AP are semilinear. However, we prove that this is not the case, and in the process we establish that the same is true for the variety SemCanRL of semilinear cancellative residuated lattices. In addition, we prove that the variety whose members have a distributive lattice reduct and satisfy the identity x(y ∧ z)w ≈ xyw ∧ xzw also fails the AP.

An MV-algebra (equivalently, a lattice-ordered Abelian group with a distinguished order unit) is strongly semisimple if all of its quotients modulo finitely generated congruences are semisimple. All MV-algebras satisfy a Chinese Remainder Theorem, as was first shown by Keimel four decades ago in the context of lattice-groups. In this note we prove that the Chinese Remainder Theorem admits a considerable strengthening for strongly semisimple structures.

Every semifield of non-zero characteristic is either a field of prime characteristic or a semifield of characteristic 1. Semifields of characteristic 1 are equivalent to abelian lattice-ordered groups. It is proved that such a semifield A is algebraically closed if and only if the pure equations x n = a and certain quadratic equations are solvable in A. Using a sheaf representation for z-projectable abelian l-groups on the co-Zariski space of minimal primes, a sheaf-theoretic characterization of algebraic closedness in characteristic 1 is obtained. Concerning the solvability of quadratic equations, the criterion consists in a topological condition for the base space. The results are built upon an analysis of rational functions in characteristic 1. While polynomials satisfy the “fundamental theorem of algebra”, the multiplicative structure of rational functions is determined by means of “divisors”, extracted from the additive structure of A modulo parallelogram identities.

Some results on coherence in probabilistic and in possibilistic frameworks are presented in order to deal with nonmonotonic reasoning. Moreover, we extend these results to conditional decomposable measures. We deal with entailment and prove that it satisfies the axiomatization of System P by referring to conditional necessities or to specific conditional decomposable measures (which include conditional probability). Finally, we study some aspects concerning a notion of irrelevance.

Many-valued binary relations are considered, taking values in a complete lattice, possibly equipped with additional operations that characterize extended-order algebras. The functionality of such relations is defined, by means of properties such as univocality and totality, from three different perspectives, namely from the viewpoint of α-cuts, of composition and of powerset operators, respectively. Functional many-valued relations are generalizations of functions and special care is devoted to characterize those which are actually functions. Relationships between the involved concepts are discussed with several results and examples.