Real Unit Interval Research Articles

The Variety Generated by Perfect BL -Algebras: an Algebraic Approach in a Fuzzy Logic Setting

i>BL-algebras are the Lindenbaum algebras of the propositional calculus coming from the continuous triangular norms and their residua in the real unit interval. Any i>BL-algebra is a subdirect product of local (linear) i>BL-algebras. A local i>BL-algebra is either locally finite (and hence an i>MV-algebra) or perfect or peculiar. Here we study extensively perfect i>BL-algebras characterizing, with a finite scheme of equations, the generated variety. We first establish some results for general i>BL-algebras, afterwards the variety is studied in detail. All the results are parallel to those ones already existing in the theory of perfect i>MV-algebras, but these results must be reformulated and reproved in a different way, because the axioms of i>BL-algebras are obviously weaker than those for i>MV-algebras.

Fuzzy Logic: Misconceptions and Clarifications

Some commonly accepted statements concerning the basic fuzzy logic proposed by Lotfi Zadeh in 1965, have led to suggestions that fuzzy logic is not a logic in the same sense as classical bivalent logic. Those considered herein are: fuzzy logic generates results that contradict classical logic, fuzzy logic collapses to classical logic, there can be no proof theory for fuzzy logic, fuzzy logic is inconsistent, fuzzy logic produces results that no human can accept, fuzzy logic is not proof-theoretic complete, fuzzy logic is too complex for practical use, and, finally, fuzzy logic is not needed. It is either proved or argued herein that all of the these statements are false and are, hence, misconceptions. A fuzzy logic with truth values specified as subintervals of the real unit interval [0.0, 1.0] is introduced. Proofs of the correctness, consistency, and proof theoretic completeness of the truth interval fuzzy logic are either summarized or cited. It is concluded that fuzzy logics deserve the accolade of logic to the same degree that the term applies to classical logics.

A new axiomatization for involutive monoidal t-norm-based logic

On the real unit interval, the notion of a Girard monoid coincides with the notion of a t-norm-based residuated lattice with strong induced negation. A geometrical approach toward these Girard monoids, based on the notion of rotation invariance, is turned in an adequate axiomatization for the involutive monoidal t-norm-based residuated logic.

Systems of ordinal fuzzy logic with application to preference modelling

In this paper, we survey several many-valued propositional logics in which the truth-functions (in the real unit interval [0,1]) of their connectives are definable only from the natural ordering of the scale, without using any richer algebraic structure. In particular we describe a complete notion of proof for weighted formulas of the implication-free fragment of Gödel logic with involution. The usefulness of this logical framework for the formalization of ordinal preference modelling is also shown.

Lattice Fully Orderable Groups

Let Ω be a linearly ordered set, A(Ω) be the group of all order automorphisms of Ω, and L(Ω) be a normal subgroup of A(Ω) consisting of all automorphisms whose support is bounded above. We argue to show that, for every linearly ordered set Ω such that: (1) A(Ω) is an o-2-transitive group, and (2) Ω contains a countable unbounded sequence of elements, the simple group A(Ω)/L(Ω) has exactly two maximal and two minimal non-trivial (mutually inverse) partial orders, and that every partial order of A(Ω)/L(Ω) extends to a lattice one (Thm. 2.1). It is proved that every lattice-orderable group is isomorphically embeddable in a simple lattice fully orderable group (Thm. 2.2). We also state that some quotient groups of Dlab groups of the real line and unit interval are lattice fully orderable (Thms. 3.1 and 3.2).

A fuzzy logic with interval truth values

The Kenevan Truth Interval Fuzzy Logic, in which truth values of propositions are represented as subintervals of the real unit interval that contain the single truth value rather than the truth value itself, is described herein. Inference rules are presented and proven to be correct, consistent, and as strong as possible. The classical propositional logic is shown to be a special case of the truth interval fuzzy logic in which the truth intervals are restricted to [0.0,0.0] and [1.0,1.0]. If the logic is restricted to propositions and formulas whose truth intervals are contained in the interval [x 0,x 1], 0.5<x 0⩽x 1⩽1.0 , or its negated equivalent, the full logic is reduced to the Zadeh's dispositional logic. A truth interval refinement proof method is described along with proof methods based on a fuzzy analog of the classical concept of logical equivalence for the dispositional logic. The latter are proven to be complete and sound. The computational complexity of the truth interval refinement proof method is shown to be Θ(2 3∗N) ; that of the proof methods based on fuzzy logical consequence, Θ(2 4∗N) ; and all of the proof methods are shown to be decidable and fair. A predicate truth interval fuzzy logic is also introduced.

THE SUBSPACE PROBLEM IN THE TRADITIONAL POINT-SET CONTEXT OF FUZZY TOPOLOGY

We consider old and new categories of fuzzy topological spaces where objects associated to arbitrary fuzzy sets are considered and fuzzy topological subspaces are characterized as particular subobjects. Also fuzzy homeomorphisms are characterized in the considered categories. We unify notation and language and compare the different frameworks in the traditional point-set context. We use only the real unit interval as a lattice in order to define fuzzy sets.

CoproductMV-Algebras, Nonstandard Reals, and Riesz Spaces

Up to categorical equivalence,MV-algebras are unit intervals of abelian lattice-ordered groups (for short,l-groups) with strong unit. While the property of being a strong unit is not even definable in first-order logic,MV-algebras are definable by a few simple equations. Accordingly, such notions as ideals and coproducts are definable for anyMV-algebraAas particular cases of the general algebraic notions. The radical RadAis the intersection of all maximal ideals ofA. AnMV-algebraAis said to be local iff it has a unique maximal ideal. Then, by Hoelder's theorem, the quotientA/RadAis isomorphic to a subalgebra of the real unit interval [0,1]. Using nonstandard real numbers we give a concrete representation of those totally orderedMV-algebrasAwhich are isomorphic to the coproduct ofA/RadAand 〈RadA〉, the latter denoting the subalgebra ofAgenerated by its radical. As an application, using several categorical equivalences we describe theMV-algebraic counterparts of Riesz spaces, also known as vector lattices.

Semi-metrics, closure spaces and digital topology

We investigate certain generalized topological structures, with the aim of showing that they can provide a suitable framework within which to compare various approaches to digital topology (including tolerance geometry) with “ordinary” topology. Within an appropriate category of these structures, ordinary spaces arise as inverse limits of digital spaces. In the first instance, the structures can be taken to be (a slight generalization of) C̆ech closure spaces. The structures of this type which seem to be the most useful are those which can be realized as topological graphs (see Section 3). Domain equations for the (real) unit interval provide our main detailed application (Section 4). Another application area which seems promising is that of modal semantics.

Extensions of real-valued difference posets

The difference poset of real-valued functions (fuzzy sets) is investigated. The difference posets on some subsets of the unit real interval [0, 1] are characterized. It is shown that although it is not always possible to represent the difference by some generator, for dense subsets such a representation exists and is unique.

Approximations and logic.

Approximations and logic.

Random sequences in generalized Cantor sets

This article presents a fast algorithm for generating random points in the finitely and infinitely defined generalized Cantor sets in the unit real interval.

L‐Fuzzy Contiguity Relations and L‐Fuzzy Closure Operators in the Case of Completely Distributive, Complete Lattices L

AbstractIn the case of completely distributive, complete lattices L the relations between L‐fuzzy contiguity relations and L‐fuzzy closure operators are investigated. In addition, if L is the real unit interval, L‐fuzzy contiguity relations are represented by means of VD‐contiguity relations, PT spaces and random VD‐contiguity relations respectively.

Measures of graphs on the reals

This paper studies measure properties of graphs with infinitely many vertices. Let [0, 1] denote the real unit interval, and 7 be the collection of bijections taking [0, 1] onto itself. Given a graph G = ([0, 1], E) and f e , define the f-representation of G to be the set Ef = { (f(x), f(y)): x, y E [0, 1] and (x, y) E E}. Let ,u be 2-dimensional Lebesgue measure. Define the mneasure spectrum of G to be the set M(G) = {m E [0, 1]: 3f E 7 with Ef measurable and jtEf = mn} . Our main result characterizes those subsets of reals that are the measure spectra of graphs.

Fuzzy stone-Čech-type compactifications

In this paper we investigate the fuzzy sets with lattice (precisely, fuzzy lattice) values. Some new global properties on the weakly induced spaces and induced spaces are obtained. Based on these results, we take the induced space of real unit interval I as new basic space and establish a reasonable theory of Stone-Čech compactifications via fuzzy imbedding theory [5] and N-compactness [10, 12]. For so-called T 3 1 2 ∗ and weakly induced spaces, its Stone-Čech-type compactification always exists. We give a counterexample to show that the previous Stone-Čech compactification [6] is not the largest T 2 compactification of a given space. Compared with this, we now have a positive results for the new Stone-Čech-type compactification. Moreover, although the preorder among the compactifications for a given space does not generally satisfy the anti-symmetric law, we fortunately establish uniqueness of the largest T 2-type compactification among the compactifications for a given space.

Structure of completely distributive complete lattices

Every completely distributive complete lattice is a subdirect product of copies of the lattice {0, 1} and the real unit interval.

Inductive inference of approximations

In this paper we investigate inductive inference identification criteria which permit infinitely many errors in explanations, but which require that the “density” of these errors be no more than a certain, prespectified amount. We introduce three hierarchies of such criteria, each of which has the same order type as the real unit interval. These three hierarchies are progressively more strict in the way they measure density of errors of explanations. The strictest of the three turns out to have all of its members, save one, incomparable to the identification criterion which permits finitely many errors in explanations.

Compactness properties in the fuzzy real line

It is the purpose of this paper to give the solution to a number of open questions in [14], concerning the compactness properties of the fuzzy real line and the fuzzy real unit interval. However using the model of the fuzzy real line which we already proposed in [9, 12] we are able to describe all compactness properties not only of the spaces mentioned above but actually of several large classes of subspaces of the fuzzy real line; and thus the answers to the questions in [14] follow as easy consequences of our general results.

Boolean embeddings of orthomodular sets and quantum logic

By a “quantum logic” we mean a pair F, P where P is a set and F is a set of functions from P to the closed real unit interval satisfying three postulates which we describe in intuitive terms here. Cf. [2], [4], [7]. P may be interpreted as the set of events and F the set of states of a “physical system”, and f(x) then becomes the probability of occurrence of the event x in the state f. Since the outcome of an experiment is an estimate for some f(x), or a collection of such estimates, it is natural to identify events which cannot be distinguished by experiment.