34 research outputs found
Some properties of state filters in state residuated lattices
summary:We consider properties of state filters of state residuated lattices and prove that for every state filter of a state residuated lattice : \begin {itemize} \item [(1)] is obstinate ; \item [(2)] is primary is a state local residuated lattice; \end {itemize} and that every g-state residuated lattice is a subdirect product of , where is a prime state filter of . \endgraf Moreover, we show that the quotient MTL-algebra of a state residuated lattice by a state prime filter is not always totally ordered, although the quotient MTL-algebra by a prime filter is totally ordered
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
Lattice-theoretic properties of algebras of logic
Abstract In the theory of lattice-ordered groups, there are interesting examples of properties -such as projectability -that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semilinear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a, 1] is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety
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
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