In a classical paper [15] V. Glivenko showed that a proposition is classically demonstrable if and only if its double negation is intuitionistically demonstrable. This result has an algebraic formulation: the double negation is a homomorphism from each Heyting algebra onto the Boolean algebra of its regular elements. Versions of both the logical and algebraic formulations of Glivenko’s theorem, adapted to other systems of logics and to algebras not necessarily related to logic can be found in the literature (see [2, (...) 9, 8, 14] and [13, 7, 14]). The aim of this paper is to offer a general frame for studying both logical and algebraic generalizations of Glivenko’s theorem. We give abstract formulations for quasivarieties of algebras and for equivalential and algebraizable deductive systems and both formulations are compared when the quasivariety and the deductive system are related. We also analyse Glivenko’s theorem for compatible expansions of both cases. (shrink)

IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation (...) to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it. (shrink)

It is well known that classical propositional logic can be interpreted in intuitionistic propositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part (...) of the paper, we also discuss some extended forms of the Kolmogorov translation and we compare it to the Glivenko translation. (shrink)

Let |$\textbf{A}$| be a |$BCK$|-algebra and |$f:A^{k}\rightarrow A$| a function. The main goal of this article is to give a necessary and sufficient condition for |$f$| to be compatible with respect to every relative congruence of |$\textbf{A}$|⁠. We extend this result in some related algebras, as e.g. in pocrims.

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class (...) of integral commutative bounded residuated lattices and to the involutive case. (shrink)

Let ${\mathbb{BRL}}$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety ${\mathbb{V}}$ of ${\mathbb{BRL}}$ is a unary term t in the language of bounded residuated lattices such that for every ${{\bf A} \in \mathbb{V}, t^{A}}$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there is (...) a variety ${\mathbb{V}_{t} \subsetneq \mathbb{BRL}}$ such that a variety ${\mathbb{V} \subsetneq \mathbb{BRL}}$ admits the unary term t as a Boolean retraction term if and only if ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ . Moreover, the equation s(x) = t(x) holds in ${\mathbb{V}_{s} \cap \mathbb{V}_{t}}$ . The radical of ${{\bf A} \in \mathbb{BRL}}$ , with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ is associated a variety ${\mathbb{V}^{r}}$ formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in ${\mathbb{V}}$ . Each free algebra in such ${\mathbb{V}}$ is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety ${\mathbb{V}^{r}}$ , also with the same number of free generators.A hierarchy of subvarieties of ${\mathbb{BRL}}$ admitting Boolean retraction terms is exhibited. (shrink)

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We include examples (...) to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

In this paper we prove that the free pseudocomplemented residuated lattices are decomposable if and only if they are Stone, i.e., if and only if they satisfy the identity ¬ x ∨ ¬¬ x = 1. Some applications are given.

In the present paper we give a description of the free algebra over an arbitrary set of generators in the variety of nilpotent minimum algebras. Such description is given in terms of a weak Boolean product of directly indecomposable algebras over the Boolean space corresponding to the Boolean subalgebra of the free NM-algebra.

This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these asvariant semanticsand present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we demonstrate that (...) failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting. (shrink)