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.

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...

On the real unit interval, the notion of a Girard monoid coincides with the notion of a t-norm-based residuated lattice with strong induced negation. A geometrical approach toward these Girard monoids, based on the notion of rotation invariance, is turned in an adequate axiomatization for the Involutive Monoidal T-norm-based residuated Logic (IMTL).

Lattice-ordered monoids are important backgrounds and algebraic foundations of residuum in general. The t-norm based lattices are investigated widely in fuzzy models, but in recent time while researching new approximate reasoning methods in soft computing based models and fuzzy models, the investigations are focused on new types of operators, like uninorms. It is necessary to find and define co...

Different notions of coherence and consistence have been proposed in the literature on fuzzy systems. In this work we focus on the relationship between some of the approaches developed, on the one hand, based of residuated lattices and, on the other hand, based on the theory of bilattices.

Sugeno integrals and their particular cases such as weighted minimum and maximum have been used in multiple-criteria aggregation when the evaluation scale is qualitative. This paper proposes two new variants of weighted minimum and maximum, where the criteria weights have a role of tolerance threshold. These variants require the use of a residuated structure, equipped with an involutive negatio...

A residuated lattice is an ordered algebraic structure L = 〈L,∧,∨, · , e, \ , / 〉 such that 〈L,∧,∨〉 is a lattice, 〈L, ·, e〉 is a monoid, and \ and / are binary operations for which the equivalences a · b ≤ c ⇐⇒ a ≤ c/b ⇐⇒ b ≤ a\c hold for all a, b, c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as “dividing” on the right...

In this article we prove a set of preservation properties of the reticulation functor for residuated lattices (for instance preservation of subalgebras, finite direct products, inductive limits, Boolean powers) and we transfer certain properties between bounded distributive lattices and residuated lattices through the reticulation, focusing on Stone, strongly Stone and m-Stone algebras. 2000 Ma...