5 research outputs found
Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics
This paper focuses on the issue of how generalizations of continuous and left-continuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continuous and left-continuous t-norms respectively. In both cases, the underlying basic structure is that of linearly ordered residuated lattices. Although the residuation property is equivalent to left-continuity in t-norms, continuous t-norms have received much more attention due to their simpler structure. We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains. In particular we show the good behavior of BL-algebras over a finite or complete chain, and discuss the partial knowledge of rational BL-chains. Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains. The unsolved problem of describing the structure of left-continuous t-norms is presented together with a fistful of construction-decomposition techniques that apply to some distinguished families of t-norms and, finally, we discuss the situation in the general study of MTL-chains as a natural generalization of left-continuous t-norms
The General Apple Property and Boolean terms in Integral Bounded Residuated Lattice-ordered Commutative Monoids
In this paper we give equational presentations of the varieties of {\em
integral bounded residuated lattice-ordered commutative monoids} (bounded
residuated lattices for short) satisfying the \emph{General Apple Property}
(GAP), that is, varieties in which all of its directly indecomposable members
are local. This characterization is given by means of Boolean terms: \emph{A
variety of \brl s has GAP iff there is an unary term such
that satisfies the equations and
, for some }. Using this
characterization, we show that for any variety of bounded
residuated lattice satisfying GAP there is such that the equation
holds in , that is, . As a consequence we improve Theorem 5.7 of
\cite{CT12}, showing in theorem that a\emph{ variety of \brls\ has Boolean
retraction term if and only if there is such that it satisfies the
equation .} We also see that in Bounded
residuated lattices GAP is equivalent to Boolean lifting property (BLP) and so,
it is equivalent to quasi-local property (in the sense of \cite{GLM12}).
Finally, we prove that a variety of \brl s has GAP and its semisimple members
form a variety if and only if there exists an unary term which is
simultaneously Boolean and radical for this variety.Comment: 25 pages, 1 figure, 2 table
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
Factor congruences in BCK-algebras
In this paper, we characterize factor congruences in the quasivariety of BCK-algebras. As an application we prove that the free algebra over an infinite set of generators is indecomposable in any subvariety of BCK-algebras. We also study the decomposability of free algebras in the variety of hoop residuation algebras (HBCK) and its subvarieties. We prove that free algebras in a non k-potent subvariety of HBCK are indecomposable while finitely generated free algebras in k-potent subvarieties have a unique non-trivial decomposition into a direct product of two factors, and one of them is the two-element implication algebra.Fil: Abad, Manuel. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: DÃaz Varela, José Patricio. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - BahÃa Blanca. Instituto de Matemática BahÃa Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática BahÃa Blanca; Argentin