Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate (...) for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens. (shrink)

Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ→˜φ)) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the (...) variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras. (shrink)

We show that the class of all isomorphic images of Boolean Products of members of SR [1] is the class of all archimedean W-algebras. We obtain this result from the characterization of W-algebras which are isomorphic images of Boolean Products of CW-algebras.

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)

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)

We give a presentation of Post algebras of ordern+1 (n1) asn+1 bounded Wajsberg algebras with an additional constant, and we show that a Wajsberg algebra admits a P-algebra reduct if and only if it isn+1 bounded.

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 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 this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ3 ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x 3) ∨ x ≈ ⊤, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally (...) ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms. (shrink)

Lattice BCK logic is the expansion of the well known Meredith implicational logic BCK expanded with lattice conjunction and disjunction. Although its natural axiomatization has three rules named modus ponens, ∨‐rule and ∧‐rule, we show that we can give an equivalent presentation with just modus ponens and ∧‐rule, however it is impossible to obtain an equivalent presentation with modus ponens as unique rule. In this paper we study and characterize all axiomatic extensions of lattice BCK logic with modus ponens as (...) unique rule. We obtain an infinite chain of proper axiomatic extensions with this property. Moreover, we prove that there is no weakest axiomatic extension of Lattice BCK‐logic admitting modus ponens as unique rule. (shrink)

In this paper we show that the quasivariety generated by an infinite simple MV-algebra only depends on the rationals which it contains. We extend this property to arbitrary families of simple MV-algebras.

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)

In this paper we characterize the MV-algebras containing as subalgebras Post algebras of finitely many orders. For this we study cyclic elements in MV-algebras which are the generators of the fundamental chain of the Post algebras.

In this paper we see that the answer of this question is affirmative. We prove this for Dco-algebras and as special case we obtain the result for Positive Implication algebras. First we give, without proof, the properties of Dco-algebras and S-algebras and their connection with Positive Implication algebras and Implication algebras. These results can be found in [T] and [IT].

In any variety of bounded integral residuated lattice-ordered commutative monoids the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the relationship with (...) the property “to have radical term”, especially for k-radical varieties, and for the hierarchy of varieties k>0 defined in Cignoli and Torrens. (shrink)