The aim of the present work is to study the  $F$-transform over a generalized residuated lattice.  We discuss the properties that are common with the $F$-transform over a residuated lattice. We show that the $F^{uparrow}$-transform can be used in establishing a fuzzy (pre)order on the set of fuzzy sets.

In this paper, we define the notions of fuzzy congruence relations and fuzzy convex subalgebras on a commutative residuated lattice and we obtain some related results. In particular, we will show that there exists a one to one correspondence between the set of all fuzzy congruence relations and the set of all fuzzy convex subalgebras on a commutative residuated lattice. Then we study fuzzy...

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the Π1−comp...

We study interior operators from the point of view of fuzzy set theory. The present approach generalizes the particular cases studied previously in the literature in two aspects. First, we use complete residuated lattices as structures of truth values generalizing thus several important cases like the classical Boolean case, (left-)continuous t-norms, MV-algebras, BL-algebras, etc. Second, and ...

It is quite well understood that propositional logics are tightly connected to ordered algebras via algebraic completeness, and because of this connection proof theory is often useful in the algebraic context too. A prominent example is that one deductively proves the interpolation theorem for a given logic in order to derive the algebraic amalgamation property for the corresponding variety as ...

Residuated structures are important lattice-ordered algebras both for mathematics and for logics; in particular, the development of lattice-valued mathematics and related non-classical logics is based on a multitude of lattice-ordered structures that suit for many-valued reasoning under uncertainty and vagueness. Extended-order algebras, introduced in [10] and further developed in [1], give an ...

The study of the Gentzen system Gew determined by the well known sequent calculus FLew [Ono98, Ono03c] is interesting for the study of the substructural aspects of t-norm based logics [Háj98, EG01]. In [BGV05] we studied the 〈∨, ∗,¬, 0, 1〉 and the 〈∨,∧, ∗,¬, 0, 1〉-fragments of this Gentzen system and the same fragments of the logic of residuated lattices IPC∗\c. In this paper we continue the st...

The logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [Höh95], intuitionistic logic without contraction [AV00], HBCK [OK85] (nowadays called FLew by Ono), etc. In this paper 1 we study the 〈∨, ∗,¬, 0, 1〉-fragment and the 〈∨,∧, ∗,¬, 0, 1〉-fragment of the logical systems associated with residuated lattices, both from the p...

As is well-known, residuated lattices (RLs) on the unit interval correspond to leftcontinuous t-norms. Thus far, a similar characterization has not been found for RLs on the set of intervals of [0,1], or more generally, of a bounded lattice L. In this paper, we show that the open problem can be solved if it is restricted, making only a few simple and intuitive assumptions, to the class of inter...

In this article, we propose an axiomatic system for fuzzy “cardinality” measures (referred to as fuzzy c-measures for short) assigning to each finite fuzzy set a generalized cardinal that expresses the number of elements that the fuzzy set contains. The system generalizes an axiomatic system introduced by J. Casasnovas and J. Torrens (2003). We show that each fuzzy c-measure is determined by tw...