In the present paper, which is a sequel to [20, 4, 12], we investigate further the structure theory of quasi-MV algebras and √ quasi-MV algebras. In particular: we provide a new representation of arbitrary √ qMV algebras in terms of √ qMV algebras arising out of their MV* term subreducts of regular elements; we investigate in greater detail the structure of the lattice of √ qMV varieties, proving that it is uncountable, providing equational bases for some of its members, as well as analysing a number of slices of special interest; we show that the variety of √ qMV algebras has the amalgamation property; we provide an axiomatisation of the 1-assertional logic of √ qMV algebras; lastly, we reconsider the correspondence between Cartesian √ qMV algebras and a category of Abelian lattice-ordered groups with operators first addressed in [10].

Kronecker called his programme of arithmetization “General Arithmetic” (Allgemeine Arithmetik). In his view, arithmetic is the building block of the whole edifice of mathematics. The aim of this paper is to show that Kronecker’s arithmetical philosophy and mathematical practice have exerted a permanent influence on a long tradition of mathematicians from Hilbert to Weil, Grothendieck and Langlands. The conclusion hints at a constructivist finitist stance in contemporary mathematical logic, especially proof theory, beyond Hilbert’s programme of finitist foundations which can be seen as the continuation of Kronecker’s arithmetization programme by metamathematical or logical means. It is finally argued that the introduction of higher-degree polynomials by Kronecker inspired Hilbert’s notion of functionals, which in turn influenced G¨odel’s functional Dialectica interpretation for his intuitionistic proof of the consistency of arithmetic.

The paper studies the logics which algebraic semantics comprises of the Hilbert algebras endowed with additional operations - the regular algebras. With any finite subdirectly irreducible regular algebra one can associate a Jankov formula. In its turn, the Jankov formulas can be used as anti-axioms for a refutation system. It is proven that a logic has a complete refutation system based on Jankov formulas if and only if this logic enjoys finite model property. Also, such a refutation system is finite, that is, it contains a finite number of axioms and anti-axioms, if and and only if the logic is tabular.

In this paper we shall study some extensions of the semilattice based deductive systems S (N) and S (N, 1), where N is the variety of bounded distributive lattices with a negation operator. We shall prove that S (N) and S (N, 1) are the deductive systems generated by the local consequence relation and the global consequence relation associated with ¬-frames, respectively. Using algebraic and relational methods we will prove that S (N) and some of its extensions are canonical and frame complete.

A tableau procedure that tests bisimulation invariance of a given first-order formula, and therefore tests if that formula is equivalent to the standard translation of some modal formula, is presented. The test is sound and complete: a given formula is bisimulation invariant if and only if there is a closed tableau for that formula. The test generally does not terminate, but it does if a given formula is bisimulation invariant, i.e., the test is positive.