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$BL$-algebras, introduced by P. Hájek, form an algebraic counterpart of the basic fuzzy logic. In the paper it is shown that $BL$-algebras are the duals of bounded representable $DRl$-monoids. This duality enables us to describe some structure properties of $BL$-algebras.

The concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters. In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework. We thus recapture theorems already known in the literature, as well as new ones. As a by-product we introduce the class of pre-BCK algebras.

We present a groupoid which can be converted into a Boolean algebra with respect to term operations. Also conversely, every Boolean algebra can be reached in this way.

Pseudo BL-algebras are a noncommutative extention of BL-algebras. In this paper we study good pseudo BL-algebras and consider some classes of these algebras.

In [13], an algebraic approach to the natural structure of domains of linguistic variables was introduced. In this approach, every linguistic domain can be interpreted as an algebraic structure called a hedge algebra. In this paper, a refinement structure of hedge algebras based on free distributive lattices generated by linguistic hedge operations will be examined in order to model structure of linguistic domains more properly. In solving this question, we restrict our consideration to the specific...

Let $A:=(A,\to ,1)$ be a Hilbert algebra. The monoid of all unary operations on $A$ generated by operations ${\alpha}_{p}:x\mapsto (p\to x)$, which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of $A$. This semilattice is isomorphic to the semilattice of finitely generated filters of $A$, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of $A$. Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of $A$, and the...

We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a...

We prove that the interval topology of an Archimedean atomic lattice effect algebra $E$ is Hausdorff whenever the set of all atoms of $E$ is almost orthogonal. In such a case $E$ is order continuous. If moreover $E$ is complete then order convergence of nets of elements of $E$ is topological and hence it coincides with convergence in the order topology and this topology is compact Hausdorff compatible with a uniformity induced by a separating function family on $E$ corresponding to compact and cocompact elements....