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...

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.

We introduce the notion of (L, e)-filters with fuzzy partially order e on complete residuated lattice L. We investigate (L, e)-filters induced by the family of (L, e)-filters and functions. In fact, we study the initial and final structures for the family of (L, e)-filters and functions. From this result, we define the product and co-product for the family of (L, e)-filters and functions.

Alten proved in [4] that the FEP holds for commutative residuated lattices that satisfy any knotted rule. A class of algebras K is said to have the FEP, if for every algebra A in K and every finite partial subalgebra B of A, there exists a finite algebra D in K such that B embeds into D. B is a finite partial subalgebra of A, if B is a finite subset of A and each n−ary operation f on A induces ...

We present the numbers of all non-isomorphic residuated lattices with up to 12 elements and a link to a database of these lattices. In addition, we explore various characteristics of these lattices such as the width, length, and various properties considered in the literature and provide the corresponding statistics. We also present algorithms for computing finite residuated lattices including ...

Injectives in several classes of structures associated with logic are characterized. Among the classes considered are residuated lattices, MTLalgebras, IMTL-algebras, BL-algebras, NM-algebras and bounded hoops.