The present paper introduces and studies the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋn of n-linear weakly Heyting algebras. It corresponds to the algebraic semantic of the strict implication fragment of the normal modal logic K with a generalization of the axiom that defines the linear intuitionistic logic or Dummett logic. Special attention is given to the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋ2 that generalizes the linear Heyting algebras studied in [10] and [12], and the linear Basic algebras introduced in [2].

In this paper we continue the study of the variety \ of monadic Gödel algebras. These algebras are the equivalent algebraic semantics of the S5-modal expansion of Gödel logic, which is equivalent to the one-variable monadic fragment of first-order Gödel logic. We show three families of locally finite subvarieties of \ and give their equational bases. We also introduce a topological duality for monadic Gödel algebras and, as an application of this representation theorem, we characterize congruences and give characterizations of (...) the locally finite subvarieties mentioned above by means of their dual spaces. Finally, we study some further properties of the subvariety generated by monadic Gödel chains: we present a characteristic chain for this variety, we prove that a Glivenko-type theorem holds for these algebras and we characterize free algebras over n generators. (shrink)

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...) algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed" conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives. (shrink)

A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that ([ 0,1], *, 1) is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and ([ 0,1], *, →, 1) becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras ([ 0,1], *, →, 1), where (...) * is a continuous t-norm. In this paper we investigate the structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras. (shrink)

A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras, where * is a continuous t-norm. In this paper we investigate the structure of the (...) variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras. (shrink)

A Q -Heyting algebra is an algebra of type such that is a Heyting algebra and the unary operation ∇ satisfies the conditions ∇0=0, a ∧∇ a = a , ∇=∇ a ∧∇ b and ∇=∇ a ∨∇ b , for any a , b ∈ H . This paper is devoted to the study of the subvariety QH L of linear Q -Heyting algebras. Using Priestley duality we investigate the subdirectly irreducible linear Q -Heyting algebras and, as consequences, we (...) derive some properties of the lattice of subvarieties of QH L and find equational bases for some of these subvarieties. (shrink)

The aim of this paper is to characterize varieties of Heyting algebras with decidable theory of their finite members. Actually we prove that such varieties are exactly the varieties generated by linearly ordered algebras. It contrasts to the result of Burris [2] saying that in the case of whole varieties, only trivial variety and the variety of Boolean algebras have decidable first order theories.