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, 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...

The reticulation of an algebra was first defined for commutative rings by Simmons [19] and it was extended by Belluce to non-commutative rings [3]. As for the algebras of fuzzy logics, Belluce also constructed the reticulation of an MV-algebra [2], G. Georgescu defined the reticulation of a quantale [8] and L. Leuştean made this construction for BL-algebras [13, 14]. In each of the papers cited...

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.