It is known that classical logic CL is the single maximal consistent logic over intuitionistic logic Int, which is moreover the single one even over the substructural logic FLew. On the other hand, if we consider maximal consistent logics over a weaker logic, there may be uncountablymany of them. Since the subvariety lattice of a given variety V of residuated lattices is dually isomorphic to th...

We introduce a new product bilattice construction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize ...

By a symmetric residuated lattice we understand an algebra A = (A,∨,∧, ∗,→,∼, 1, 0) such that (A,∨,∧, ∗,→, 1, 0) is a commutative integral bounded residuated lattice and the equations ∼∼ x = x and ∼ (x ∨ y) =∼ x∧ ∼ y are satisfied. The aim of the paper is to investigate properties of the unary operation ε defined by the prescription εx :=∼ x → 0. We give necessary and sufficient conditions for ...

Let L be a complete residuated lattice. Then we show that any L-preorder can be represented both by an implication-based graded inclusion as defined [1] and by a similarity-based graded inclusion as defined in [2]. Also, in accordance with a duality between [0,1]-orders and quasi-metrics, we obtain two corresponding representation theorems for quasi-metrics.

We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pa...

In this paper, we introduce the notion of L-topological spaces based on a complete bounded integral residuated lattice and discuss some properties of interior and left (right) closure operators.

The assertional logic S(BCIA) of the quasivariety of BCI-algebras (in Iseki's sense) is axiomatized, relative to pure implicational logic BCI, by the rule x, y, x → y (G) (see [1]). Alternatively, the role of (G) can be played by x x → (y → y) (1) (see [2]). The formula (x → x) → (y → y) (2) is a theorem of S(BCIA). In [2, Proposition 22] we claimed erroneously that, relative to BCI, the axiom ...

This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones only evaluated in 0 and 1. ...

We generalize the notion of complete binary relation on complete lattice to residuated lattice valued ordered sets and show its properties. Then we focus on complete fuzzy tolerances on fuzzy complete lattices and prove they are in one-to-one correspondence with extensive isotone Galois connections. Finally, we prove that fuzzy complete lattice, factorized by a complete fuzzy tolerance, is agai...

Petr Hájek identified the logic BL, that was later shown to be the logic of continuous t-norms on the unit interval, and defined the corresponding algebraic models, BL-algebras, in the context of residuated lattices. The defining characteristics of BL-algebras are representability and divisibility. In this short note we survey recent developments in the study of divisible residuated lattices an...