In this paper, we investigate two L-neighborhood systems induced by an Lfuzzy topology in a complete residuated lattice L. We study the relationships among L-fuzzy topology, L-topologies and L-neighborhood systems. Finally, we give their examples. AMS Subject Classification: 03E72, 06A15, 06F07, 54A05

Presented is a completeness theorem for fuzzy equational logic with truth values in a complete residuated lattice: Given a fuzzy set Σ of identities and an identity p ≈ q, the degree to which p ≈ q syntactically follows (is provable) from Σ equals the degree to which p ≈ q semantically follows from Σ. Pavelka style generalization of well-known Birkhoff’s theorem is therefore established.

Continuing with our general study of algebraic hyperstructures, we focus on the residuated operation in the framework of multilattices. Firstly, we recall the existing relation between filters, homomorphisms and congruences in the framework of multilattices; then, introduce the notion of residuated multilattice and further study the notion of filter, which has to be suitably modified so that th...

We introduce the concept of very true operator on an effect algebra. Although an effect algebra is only partial, we define it in the way which is in accordance with traditional definitions in residuated lattices or basic algebras. This is possible if we require monotonicity as an additional condition. We prove that very true operators on effect algebras can be characterized by means of certain ...

In this chapter we survey some developments in topological duality theory and the theory of completions for lattices with additional operations paying special attention to various classes of residuated lattices which play a central role in substructural logic. We hope this chapter will serve as an introduction and invitation to these subjects for researchers and students interested in residuate...

In this paper, at first we define the concepts of response function and accessible states of a complete residuated lattice-valued (for simplicity we write $mathcal{L}$-valued) tree automaton with a threshold $c.$ Then, related to these concepts, we prove some lemmas and theorems that are applied in considering some decision problems such as finiteness-value and emptiness-value of recognizable t...

In this paper, we define a property, trimness, for lattices. Trimness is a not-necessarily-graded generalization of distributivity; in particular, if a lattice is trim and graded, it is distributive. Trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sph...

We study a class of lattices called weak* complemented lattices which are shown to have the property that the order complex of any interval of the lattice is either contractible or homotopy equivalent to a sphere. The two main examples are lattices generated by intervals in a total order and the lattices of partitions of integers under dominance order. The proofs are done mainly using homotopy ...

In this paper we define the reticulation of a residuated lattice, prove that it has “good properties“, present two constructions for it, prove its uniqueness up to an isomorphism, define the reticulation functor and give several examples of finite residuated lattices and their reticulations.