Abstract In this work, we provide constructions, characterizations, geometrical dualities and a McNaughton theorem for non-archimedean MV-algebras, which are the semantics of super-Łukasiewicz logics introduced by Komori.

Abstract In this paper we aim at completing the study of $\sigma $-complete Riesz MV-algebras that started in Di Nola et al. (2018, J. Logic Comput., 28, 1275–1292). To do so, we discuss polynomials, algebraic geometry and dualities in the infinitary variety of such algebras. In particular, we characterize the free objects as algebras of Baire-measurable functions and we generalize two dualities, namely the Marra–Spada duality and the Gelfand duality, obtaining a duality with basically disconnected compact Hausdorff spaces and an equivalence with Rickart $C^*$-algebras.

We study commutative idempotent semirings in general, and some examples in particular. We show that the class Red of semiring reducts of MV-algebras, although axiomatized by a first order theory, is not axiomatized by a geometric theory (in the topos-theoretic sense) or a universal-existential first order theory. Then we perform comparisons between the class Red, the class of all semirings, and some so-called exotic semirings.

We focus on Riesz MV-algebras, which are MV-algebras equipped with a multiplication by numbers in the real interval [0,1]. In analogy with a work in preparation for MV-algebras by the same authors, we consider for every integer n the Riesz MV-algebra of all continuous functions from the n-th power of [0,1] to [0,1] and the Riesz MV-subalgebras thereof. In particular we study the Riesz MV-subalgebras isomorphic to free Riesz MV-algebras with finitely many generators, possibly different from the usual linear models given by what we call Riesz-McNaughton functions (and which generalize McNaughton functions used in the case of MV-algebras). In doing this we characterise zerosets of Riesz-McNaughton functions by means of polyhedra, and we extend to Riesz MV-algebras a duality for MV-algebras exposed in a paper by Marra and Spada.

In this article, first we generalize from the MV algebra [0,1] to an arbitrary MV algebra A the well-known Galois connection (V,I) between the powerset of each power of [0,1] and the powerset of the corresponding free MV algebra. Then, in analogy with the Nullstellensatz of classical algebraic geometry, we study the closure operators obtained by composing the functors V and I.

Abstract In this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

We consider the μ-calculus over graphs where the accessibility relation is an equivalence (S5-graphs). We show that the vectorial μ-calculus model checking problem over arbitrary graphs reduces to the vectorial, existential μ-calculus model checking problem over S5 graphs. Moreover, we give a proof that satisfiability of μ-calculus in S5 is NP-complete, and by using S5 graphs we give a new proof that the satisfiability problem of the existential μ-calculus is also NP-complete. Finally we prove that on multimodal S5, in contrast with the monomodal case, the fixpoint hierarchy of the μ-calculus is infinite and the finite model property fails.

We consider the hierarchy of the modal μ-calculus over reflexive and symmetric graphs and show that in this class the modal μ-calculus hierarchy is infinite. In the proof, a parity game over a tree is transformed into a equivalent parity game where Duplicator, when playing over the reflexive and symmetric closure of the tree, will never use loops or back edges.

We consider three basic questions regarding the extension of modal logic with a special kind of propositional quantifiers, known as bisimulation quantifiers, over arbitrary classes of frames: bisimulation invariance, uniform interpolation, and expressive power. In particular: – we discuss the relation between bisimulation invariance of bisimulation quantifiers and the semantical notion of amalgamation of the class of frames; – we consider a strong form of interpolation, uniform interpolation, and its relation with the closure under bisimulation quantifiers; – we compare bisimulation quantifiers logic with the better known extension of modal logic with extremal fixed points.

This paper is the journal version of Lenzi (2006). We consider the positive set theory with equality described these and propose a candidate model.

Closed monadic /spl Sigma//sub 1/, as proposed in (Ajtai et al., 1998), is the existential monadic second order logic where alternation between existential monadic second order quantifiers and first order quantifiers is allowed. Despite some effort very little is known about the expressive power of this logic on finite structures. We construct a tree automaton which exactly characterizes closed monadic /spl Sigma//sub 1/ on the Rabin tree and give a full analysis of the expressive power of closed monadic /spl Sigma//sub 1/ in this context. In particular we prove that the hierarchy inside closed monadic /spl Sigma//sub 1/, defined by the number of alternations between blocks of first order quantifiers and blocks of existential monadic second order quantifiers collapses, on the infinite tree, to the level 2.

As is already known from the work of D. Janin & I. Walukiewicz (1996), the mu-calculus is as expressive as the bisimulation-invariant fragment of monadic second-order logic. In this paper, we relate the expressiveness of levels of the fixpoint alternation depth hierarchy of the mu-calculus (the mu-calculus hierarchy) with the expressiveness of the bisimulation-invariant fragment of levels of the monadic quantifiers alternation-depth hierarchy (the monadic hierarchy). From J. van Benthem's (1976) results, we know already that the fixpoint free fragment of the mu-calculus (i.e. polymodal logic) is as expressive as the bisimulation-invariant fragment of monadic /spl Sigma//sub 0/ (i.e. first-order logic). We show that the /spl nu/-level of the mu-calculus hierarchy is as expressive as the bisimulation-invariant fragment of monadic /spl Sigma//sub 1/ and that the /spl nu//spl mu/-level of the mu-calculus hierarchy is as expressive as the bisimulation-invariant fragment of monadic /spl Sigma//sub 2/, and we show that no other level /spl Sigma//sub k/ (for k>2) of the monadic hierarchy can be related similarly with any other level of the mu-calculus hierarchy. The possible inclusion of all the mu-calculus in some level /spl Sigma//sub k/ of the monadic hierarchy, for some k>2, is also discussed.