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, 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.

For a function f of one preordered set X to another Y , we shall establish several consequences of the following two definitions: (a) f is increasingly φ–regular, for some function φ of X to itself, if for any x1 , x2 ∈ X we have x1 ≤ φ (x2) if and only if f (x1) ≤ f (x2); (b) f is increasingly g–normal, for some function g of Y to X , if for any x ∈ X and y ∈ Y we have f (x) ≤ y if and only if...

We describe properties of compositions of isotone bonds between L-fuzzy contexts over different complete residuated lattices and we show that L-fuzzy contexts as objects and isotone bonds as arrows form a category.

In the present paper we introduce and study L-pre-T0-, L-pre-T1-, L-pre-T2 (L-pre-Hausdorff)-, L-pre-T3 (L-preregularity)-, L-pre-T4 (L-pre-normality)-, L-pre-strong-T3-, L-pre-strong-T4-, L-pre-R0-, L-pre-R1-separation axioms in (2, L)-topologies where L is a complete residuated lattice. Sometimes we need more conditions on L such as the completely distributive law or that the ”∧” is distribut...

in this paper we extend the notion of  degrees of membership and non-membership of intuitionistic fuzzy sets to lattices and  introduce a residuated lattice with appropriate operations to serve as semantics of intuitionistic fuzzy logic. it would be a step forward to find an algebraic counterpart for intuitionistic fuzzy logic. we give the main properties of the operations defined and prove som...

We look at lower semilattice-ordered residuated semigroups and, in particular, the representable ones, i.e., those that are isomorphic to algebras of binary relations. We will evaluate expressions (terms, sequents, equations, quasi-equations) in representable algebras and give finite axiomatizations for several notions of validity. These results will be applied in the context of substructural l...