Residuated lattices form the algebraic counterpart of substructural logics [4]. Rotation construction for t-norms has been introduced in [9] and has been generalized to arbitrary residuated posets in [7]. The reader is referred also to [13, 6, 5]. The rotationannihilation construction has been introduced in [10, 7]. The rotation construction has proved fundamental in the structural description ...

We investigate the variety of residuated lattices with a commutative and idempotent monoid reduct. A residuated lattice is an algebra A = (A,∨,∧, ·, e, /, \) such that (A,∨,∧) is a lattice, (A, ·, e) is a monoid and for every a, b, c ∈ A ab ≤ c ⇔ a ≤ c/b ⇔ b ≤ a\c. The last condition is equivalent to the fact that (A,∨,∧, ·, e) is a lattice-ordered monoid and for every a, b ∈ A there is a great...

In this paper, the concept of intuitionistic fuzzy soft sets is applied to residuated lattices. The notion of intuitionistic fuzzy soft filters of a residuated lattice is introduced and some related properties are investigated. Then the representations of generated intuitionistic fuzzy soft filters are established. We prove that the lattice of all intuitionistic fuzzy soft filters of a residuat...

The commutative residuated lattices were first introduced by M. Ward and R.P. Dilworth as generalization of ideal lattices of rings. Non-commutative residuated lattices, called sometimes pseudo-residuated lattices, biresiduated lattices or generalized residuated lattices are algebraic counterpart of substructural logics, that is, logics which lack some of the three structural rules, namely cont...

In this paper, the concept of intuitionistic fuzzy congruence relation on a residuated lattice is introduced and its properties is studied. The relationship between intuitionistic fuzzy filters and intuitionistic fuzzy congruence relations on a residuated lattice is obtain. Then the intuitionistic fuzzy congruence relation corresponding to a given intuitionistic fuzzy filter on residuated latti...

In Section 2 replace the definition of ∗◦Q in Definition 1 by x∗◦Q y = inf{u ∗◦ v | u > x, v > y}. It is defined only if the infimum exists. Proposition 1 remains unchanged. Theorem 0. Let (X, ∗◦,→∗◦,≤) be a commutative residuated semigroup on a complete chain equipped with the order topology. Let a, b, c ∈ X be such that a = b→∗◦ c. Let (x, y) ∈ X × X be such that 1. neither x nor y equals the...

It is well known that classical propositional logic can be interpreted in intuitionistic propositional logic. In particular Glivenko’s theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko’s theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via ...

In this paper, we investigate the properties of fuzzy Galois (dual Galois, residuated, and dual residuated) connections in a complete residuated lattice L. We give their examples. In particular, we study fuzzy Galois (dual Galois, residuated, dual residuated) connections induced by L-fuzzy relations.

In the first part, we extend our results from a previous paper on factorization of residuated lattices to residuated lattices with hedges. In the second part, we show how this result can be applied to the problem of factorization of fuzzy concept lattices with hedges. Our approach is that instead of factorizing the original concept lattice with hedges we construct a new data table with fuzzy va...

The finite embeddability property (FEP) for integral, commutative residuated ordered monoids was established by W. J. Blok and C. J. van Alten in 2002. Using Higman’s finite basis theorem for divisibility orders we prove that the assumptions of commutativity and associativity are not required: the classes of integral residuated ordered monoids and integral residuated ordered groupoids have the ...