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...

The paper is a continuation of our previous paper on operators and spaces associated to matrices whose degrees are elements from a residuated lattice. The motivation for this study is to develop a calculus for such matrices which can be used in situations such as matrix decompositions. In this paper, we focus on row and column spaces, left and right ideals of matrices, and Green’s relations. We...

In the same paper they present algebraic characterizations for all these notions and investigate their relation with the usual interpolation properties. The authors also remark: “only a few properties specific to FL-algebras or residuated lattices are used in our discussion,” and they propose the study of these interpolation properties in a more general setting. The objective of the present pap...

Discretizations of the Laplacian operator on non-hypercubical lattices are discussed in a systematic approach. It is shown that order a2 errors always exist for discretizations involving only nearest neighbors. Among all lattices with the same density of lattice sites, the hypercubical lattices always have errors smaller than other lattices with the same number of spacetime dimensions. On the o...

We show that the equational theory of representable lower semilattice-ordered residuated semigroups is finitely based. We survey related results.