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...

Article history: Received 15 January 2015 Received in revised form 4 June 2015 Accepted 26 July 2015 Available online 7 August 2015

The class of all MTL-algebras is a variety, denoted MTL. Alternatively, an MTL-algebra is a representable, commutative, integral residuated lattice with a least element.

We show that the equational theory of representable lattice-ordered residuated semigroups is not finitely axiomatizable. We apply this result to the problem of completeness of substructural logics.