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.

We study finite residuated lattices with up to 11 elements. We present an algorithm for generating all non-isomorphic finite residuated lattices with a given number of elements. Furthermore, we analyze selected properties of all the lattices generated by our algorithm and present summarizing statistics.

Left-continuity of triangular norms is the characteristic property to make it a residuated lattice. Nowadays residuated lattices are subjects of intense investigation in the ÿelds of universal algebra and nonclassical logic. The recently known construction methods resulting in left-continuous triangular norms are surveyed in this paper.

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at le...

In this paper, we investigate the properties of antitone Galois connection and formal concepts. Moreover, we show that order reverse generating maps induce formal, attribute oriented and object oriented concepts on a complete residuated lattice.

Basic Logic was introduced by Petr Hájek to provide a uni ed approach to fuzzy logics, and judging by its rapid adoption in the research community, it has enjoyed considerable success in this regard. One of the reasons is that while it is a very general logic, it has elegant semantics with respect to the real unit interval, which allow for practical applications and use of standard software too...

We consider varieties of pointed lattice-ordered algebras satisfying a restricted distributivity condition and admitting a very weak implication. Examples of these varieties are ubiquitous in algebraic logic: integral or distributive residuated lattices; their {·}-free subreducts; their expansions (hence, in particular, Boolean algebras with operators and modal algebras); and varieties arising ...

In the last years the possibility to define the same algebraic structure on a non-empty interval in bounded ordered algebras were intensively studied. We provide an overview of the recent progress in this field for many types of so called quantum structures. For example, the last contribution is by I. Chajda and J. Kühr, who showed (in not yet published paper) the possibility to introduce it ev...

Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct Triangle Logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRLs. In this paper, we prove that the so-called pseudo-prelinear tri...