Interval-valued algebras and fuzzy logics

In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter

Why most papers on filters are really trivial (including this one)

The aim of this note is to show that many papers on various kinds of filters (and related concepts) in (subreducts of) residuated structures are in fact easy consequences of more general results that have been known for a long time

Why most papers on filters are really trivial (including this one)

The aim of this note is to show that many papers on various kinds of filters (and related concepts) in (subreducts of) residuated structures are in fact easy consequences of more general results that have been known for a long time

Connections between commutative rings and some algebras of logic

In this paper using the connections between some subvarieties of residuated lattices, we investigated some properties of the lattice of ideals in commutative and unitary rings. We give new characterizations for commutative rings $A$ in which $Id(A)$ is an MV-algebra, a Heyting algebra or a Boolean algebra and we establish connections between these types of rings. We are very interested in the finite case and we present summarizing statistics. We show that the lattice of ideals in a finite commutative ring of the form $A=\mathbb{Z}_{k_{1}}\times \mathbb{Z}_{k_{2}}\times ...\times \mathbb{Z}_{k_{r}},$ where $k_{i}=p_{i}^{\alpha _{i}}$ and $p_{i}$ a prime number, for all $i\in \{1,2,...,r\},$ \ is a Boolean algebra or an MV-algebra (which is not Boolean). Using this result we generate the binary block codes associated to the lattice of ideals in finite commutative rings and we present a new way to generate all (up to an isomorphism) finite MV-algebras using rings

Projectivity in (bounded) integral residuated lattices

In this paper we study projective algebras in varieties of (bounded) commutative integral residuated lattices from an algebraic (as opposed to categorical) point of view. In particular we use a well-established construction in residuated lattices: the ordinal sum. Its interaction with divisibility makes our results have a better scope in varieties of divisibile commutative integral residuated lattices, and it allows us to show that many such varieties have the property that every finitely presented algebra is projective. In particular, we obtain results on (Stonean) Heyting algebras, certain varieties of hoops, and product algebras. Moreover, we study varieties with a Boolean retraction term, showing for instance that in a variety with a Boolean retraction term all finite Boolean algebras are projective. Finally, we connect our results with the theory of Unification

Suburb en mobiliteit: zijn alle 'stedelijke gebieden' even stedelijk?

Het Ruimtelijk Structuurplan Vlaanderen (RSV) maakt een selectie van stedelijke gebieden en legt de basis voor de afbakening ervan. Het RSV onderscheidt drie basiscategorieën: drie grootstedelijke ge-bieden, een tiental regionaalstedelijke gebieden en een reeks kleinstedelijke gebieden. Het woonloca-tiebeleid dat op basis van deze selectie werd uitgestippeld, voorziet dat 60% van de nieuwe woningen in de stedelijke gebieden zou moeten terechtkomen. Een recente evaluatie van het RSV stelt vast dat een steeds groter aandeel van de nieuwe woningen inderdaad in de stedelijke gebieden wordt ge-bouwd, maar dat de nadruk duidelijk op de kleinstedelijke gebieden ligt. Een nieuwe analyse op basis van de gegevens van het Onderzoek Verplaatsingsgedrag Vlaanderen (2001) toont aan dat inwoners van kleinstedelijke gebieden bovengemiddeld lange verplaatsingen ma-ken, en dus voor veel van hun dagelijkse activiteiten wellicht in sterke mate aangewezen zijn op de vaak vrij veraf gelegen agglomeraties. Het mobiliteitsprofiel van de inwoners van suburbane gemeen-ten in de schaduw van de grotere steden is dus niet noodzakelijk minder duurzaam dan dat van de in-woners van de kleinstedelijke gebieden, terwijl de grootstedelijke en regionaalstedelijke gebieden zelf duidelijk een stuk beter scoren