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

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.

In this paper we define, inspired by ring theory, the class of maximal residuated lattices with lifting Boolean center and prove a structure theorem for them: any maximal residuated lattice with lifting Boolean center is isomorphic to a finite direct product of local residuated lattices. MSC: 06F35, 03G10.

In this paper, we study the separtion axioms T0, T1, T2 and T5/2 on topological and semitopological residuated lattices and we show that they are equivalent on topological residuated lattices. Then we prove that for every infinite cardinal number α, there exists at least one nontrivial Hausdorff topological residuated lattice of cardinality α. In the follows, we obtain some conditions on (semi)...

The theory of residuated lattices, first proposed by Ward and Dilworth [4], is formalised in Isabelle/HOL. This includes concepts of residuated functions; their adjoints and conjugates. It also contains necessary and sufficient conditions for the existence of these operations in an arbitrary lattice. The mathematical components for residuated lattices are linked to the AFP entry for relation al...