Left-continuous Triangular Norms Research Articles

On (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices

Rough sets were introduced by Pawlak as a formal tool for dealing with imprecision and uncertainty in data analysis. After that time, varieties of fuzzy generalizations of rough approximations have been investigated in the literature. In particular, generalized fuzzy rough sets determined by a triangular norm have been considered. On the other hand, binary operations & and ⊙ on complete residuated and co-residuated lattice L can be regarded as two more extensive operations than left continuous triangular norms and right continuous triangular conorms, respectively. Therefore, as a further generalization of the notion of rough sets, this paper is devoted to proposing (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices from both constructive and axiomatic approaches. In the constructive approach, we define a pair of lower and upper L-fuzzy rough approximation operators determined by ⊙ and &, respectively. Meanwhile, various classes of (⊙,&)-fuzzy rough sets are investigated. In the axiomatic approach, the axiomatic characterizations of different (⊙,&)-fuzzy rough approximation operators are studied. Furthermore, the topological properties of (⊙,&)-fuzzy rough sets are discussed, especially from the categorical point of view. At the end of this paper, we give a brief introduction to a new model of fuzzy rough sets and discuss some properties of it.

Real coextensions as a tool for constructing triangular norms

We present in this paper a universal method of constructing left-continuous triangular norms (l.-c. t-norms). The starting point is an arbitrary, possibly finite, totally ordered monoid fulfilling the conditions that are characteristic for l.-c. t-norms: commutativity, negativity, and quanticity. We show that, under suitable conditions, we can extend this structure by substituting each element for a real interval. The process can be iterated and if the final structure obtained in this way is order-isomorphic to a closed real interval, its monoidal operation can, up to isomorphism, be identified with a l.-c. t-norm.We specify the constituents needed for the construction in an explicit way. We furthermore illustrate the method on the basis of a number of examples.

Pseudovarieties of algebras with fuzzy equalities

We establish analogy of the Eilenberg–Schützenberger pseudovariety theorem in the setting of algebras with fuzzy equalities and Pavelka-style semantics using complete residuated lattices as structures of truth degrees. We show that under the assumption of countable reconstructibility, classes of finite algebras with fuzzy equalities closed under formations of subalgebras, homomorphic images, and finite direct products, are classes of finite algebras with fuzzy equalities which satisfy all but finitely many identities taken from fuzzy sets of identities which serve as graded equational theories. We also show that the assumption of countable reconstructibility is satisfied by model classes if one uses complete residuated lattices based on left-continuous triangular norms as structures of truth degrees.

适宜于展开模糊推理的两类模糊度量空间

The metric structures for measure the robustness of fuzzy reasoning are given. Based on the left-continuous triangular norms, four types of fuzzy metric spaces are constructed. Finally, it proved that the fuzzy metric spaces related to Lukasiewicz implication and Goguen implication are more suitable for fuzzy reasoning.

Continuity of left-continuous triangular norms with special associated negations

First, we show that the class of left-continuous triangular norms whose associated negations are discontinuous only at one point is not generally continuous except for the trivial case. Second, we find the strongest left-continuous triangular norm with a given associated negation and shows that there is no weakest left-continuous triangular norm with a given associated negation.

Finite RDP-algebras: duality, coproducts and logic

The variety of RDP-algebras forms the algebraic semantics of RDP-logic, the many-valued propositional logic of the revised drastic product left-continuous triangular norm and its residual. We prove a Priestley duality for finite RDP-algebras, and obtain an explicit description of coproducts of finite RDP-algebras. In this light, we give a combinatorial representation of free finitely generated RDP-algebras, which we exploit to construct normal forms, strongest deductive interpolants and most general unifiers.We prove that RDP-unification is unitary, and that the tautology problem for RDP-logic is coNP-complete.

Dual tableau for monoidal triangular norm logic MTL

Monoidal triangular norm logic MTL is the logic of left-continuous triangular norms. In the paper we present a relational formalization of the logic MTL and then we introduce relational dual tableau that can be used for verification of validity of MTL-formulas. We prove soundness and completeness of the system.

Algebra and Probability in Many-Valued Reasoning

Many-valued reasoning is a well-established domain of investigation with thriving connections to several areas of logic, mathematics, philosophy, and computer science. In May 2009, an international workshop on Algebra and Probability in Many-Valued Logics was held in Darmstadt, Germany. Established as well as younger researchers in the field gathered together to foster progress on various aspects of many-valued reasoning, with special emphasis on the topics indicated by the meeting's title. Traditionally, algebraic techniques play an important role in the investigation of manyvalued logical systems. More recently, efforts to extend the scope of the theory of probabilities to such logics are attracting much attention. As a follow-up to the workshop, we are very pleased to offer the present special issue of Studia Logica. The papers herein contained were submitted to the journal in response to the call for paper that was issued after the workshop. Some of them are related to presentations that their authors made at the meeting; others were submitted independently of the conference. In all cases, submitted papers have been reviewed, and then selected or rejected, according to the usual standards of the journal. The paper by Aguzzoli and Gerla is concerned with the logic of nilpotent minimum (NM), a schematic extension of the logic of all left-continuous triangular norms introduced by Esteva and Godo. The authors establish for the finitely generated Tarski-Lindenbaum algebras of NM a key result about generalised assignments of probabilities, known as states. Namely, they show that an appropriate axiomatisation in the style of Kolmogorov for such states provides a precise mathematical formulation of the intuitive notion of average degree of truth of a formula, where the average is computed by integrating the function expressed by the formula with respect to an appropriate measure. Results of this type had previously been proved for

T-Norm based cuts of intuitionistic fuzzy sets

We consider the system of intuitionistic fuzzy sets (IF-sets) in a universe X and study the cuts of an IF-set. Suppose a left continuous triangular norm is given. The t-norm based cut (level set) of an IF-set is defined in a way that binds the membership and nonmembership functions via the triangular norm. This is an extension of usual cuts of IF-sets. We show that the system of these cuts fulfils analogical properties as usual systems of cuts. However, it is not possible to reconstruct an IF-set from the system of t-norm based cuts.

Semi-divisible triangular norms

Semi-divisibility of left-continuous triangular norms is a weakening of the divisibility (i.e., continuity) axiom for t-norms. In this contribution we focus on the class of semi-divisible t-norms and show the following properties: Each semi-divisible t-norm with Ran(n T ) = [0, 1] is nilpotent. Semi-divisibility of an ordinal sum t-norm is determined by the corresponding property of its first component (which can be a proper t-subnorm, too). Finally, negations with finite range derived from semi-divisible t-norms are studied.

Spectral Duality for Finitely Generated Nilpotent Minimum Algebras, with Applications

We establish a categorical duality for the finitely generated Lindenbaum-Tarski algebras of propositional nilpotent minimum logic. The latter's conjunction is semantically interpreted by a left-continuous (but not continuous) triangular norm; implication is obtained through residuation. Our duality allows one to transfer to nilpotent minimum logic several known results about inutitionistic logic with the prelinearity axiom (also called Godel-Dummett logic), mutatis mutandis. We give several such applications.

Semi-divisible t-norms on discrete scales

Semi-divisibility of left-continuous triangular norms is a weakened version of the divisibility axiom. The t-norms on L = { 0 , 1 / n , 2 / n , … , 1 } which fulfil the semi-divisibility axiom are studied. The only semi-divisible t-norm with Ran ( n T ) = L is shown to correspond to the Łukasiewicz t-norm. Several examples of semi-divisible discrete t-norms are shown.

Archimedean components of triangular norms

AbstractThe Archimedean components of triangular norms (which turn the closed unit interval into anabelian, totally ordered semigroup with neutral element 1) are studied, in particular their extension to triangular norms, and some construction methods for Archimedean components are given. The triangular norms which are uniquely determined by their Archimedean components are characterized. Using ordinal sums and additive generators, new types of left-continuous triangular norms are constructed.

How to construct left-continuous triangular norms—state of the art

Left-continuity of triangular norms is the characteristic property to make it a residuated lattice. Nowadays residuated lattices are subjects of intense investigation in the fields of universal algebra and nonclassical logic. The recently known construction methods resulting in left-continuous triangular norms are surveyed in this paper.

On the existence and construction of T-transitive closures

In this paper, it is shown that any fuzzy relation R on an arbitrary universe X has a T-transitive closure. The triangular norm T involved is not subject to any conditions. This existential result can be turned into an explicit expression in the case of a left-continuous triangular norm T. Furthermore, this continuity restriction vanishes in the case of a finite universe X. In that case, the explicit expression can also be operationalized.

Structure of left-continuous triangular norms with strong induced negations. (III): Construction and decomposition

This paper is the continuation of Jenei (J. Appl. Non-Classical Logics 10 (2000) 83–92; 11 (2001) 351–366) where the rotation construction and the rotation-annihilation construction have been presented, respectively. Both constructions produce left-continuous (but not continuous) triangular norms with strong induced negations. Here, we show that these two constructions allow us to define indecomposability and that each decomposable left-continuous triangular norm with strong induced negation can be derived as result of one of the above constructions. In this way two general decomposition theorems arise. In addition, the decomposition of left-continuous triangular norms with strong induced negations can be defined in a unique way, namely, when the decomposition is done by the minimal decomposition point.

Metrics and [formula omitted]-Equalities

The relationship between metrics and T-equalities is investigated; the latter are a special case of T-equivalences, a natural generalization of the classical concept of an equivalence relation. It is shown that in the construction of metrics from T-equalities triangular norms with an additive generator play a key role. Conversely, in the construction of T-equalities from metrics this role is played by triangular norms with a continuous additive generator or, equivalently, by continuous Archimedean triangular norms. These results are then applied to the biresidual operator ET of a triangular norm T. It is shown that ET is a T-equality on [0, 1] if and only if T is left-continuous. Furthermore, it is shown that to any left-continuous triangular norm T there correspond two particular T-equalities on F(X), the class of fuzzy sets in a given universe X; one of these T-equalities is obtained from the biresidual operator ETT by means of a natural extension procedure. These T-equalities then give rise to interesting metrics on F(X).

Continuity of left-continuous triangular norms with strong induced negations and their boundary condition

First, a characterization is given for the continuity of a class of left-continuous triangular norms: it is established that a left-continuous triangular norm with strong induced negation is continuous if and only if it is strictly increasing on the domain where its value is positive. Second, we show that axioms of a left-continuous triangular norm with strong induced negation are not independent, namely, the boundary condition follows from the other axioms.

Structure of left-continuous triangular norms with strong induced negations (II) Rotation-annihilation construction

This paper is the continuation of [11] where the rotation construction of left-continuous triangular norms was presented. Here the class of triangular subnorms and a second construction, called rotation-annihilation, are introduced: Let T1 be a left-continuous triangular norm. If T1 has no zero divisors then let T2 be a left-continuous rotation invariant t-subnorm. If T1 has zero divisors then let T2 be a left-continuous rotation invariant triangular norm. From each such pair (T1, T2) the rotation-annihilation construction produces a left-continuous triangular norm with strong induced negation. An infinite number of new families of such triangular norms can be constructed in this way, and this further extends our spectrum of choice for the proper triangular norm e.g. in probabilistic (statistical) metric spaces, or for logical and set theoretical connectives in non-classical logic, or e.g. in fuzzy sets theory and its applications. On the other hand, the introduced construction brings us closer to the understanding of the structure of these operations.

Convergence of residuated operators and connective stability of non-classical logics

Some of the results of Jenei (1993) are put in a more general setting in this paper. The convergence of the residuated connectives related to a sequence of left-continuous triangular norms to the residuated connective of the limit triangular norm is studied. Two sufficient and necessary conditions are given. A derived concept, the connective stability of non-classical logics is introduced and investigated. It is established that Gödel's logic, some Girard's logics and further non-classical logics are connective stable.