41 research outputs found
Structural and universal completeness in algebra and logic
In this work we study the notions of structural and universal completeness
both from the algebraic and logical point of view. In particular, we provide
new algebraic characterizations of quasivarieties that are actively and
passively universally complete, and passively structurally complete. We apply
these general results to varieties of bounded lattices and to quasivarieties
related to substructural logics. In particular we show that a substructural
logic satisfying weakening is passively structurally complete if and only if
every classical contradiction is explosive in it. Moreover, we fully
characterize the passively structurally complete varieties of MTL-algebras,
i.e., bounded commutative integral residuated lattices generated by chains.Comment: This is a preprin
Quasivarieties of Wajsberg hoops
In this paper we deal with quasivarieties of residuated structures which form the equivalent algebraic semantics of a positive frag-
ment of some substructural logic. Our focus is mainly on varieties and quasivarieties of Wajsberg hoops, which are the equivalent
algebraic semantics of the positive fragment of Łukasiewicz many-valued logic. In particular we study the lattice of subquasivari-
eties of Wajsberg hoops and we describe completely all the subvarieties of Wajsberg hoops that are primitive. Though the treatment
is mostly algebraic in nature, there are obvious connections with the underlying logic
Expanding FLew with a Boolean connective
We expand FLew with a unary connective whose algebraic counterpart is the
operation that gives the greatest complemented element below a given argument.
We prove that the expanded logic is conservative and has the Finite Model
Property. We also prove that the corresponding expansion of the class of
residuated lattices is an equational class.Comment: 15 pages, 4 figures in Soft Computing, published online 23 July 201
Advances in the theory of μŁΠ algebras
Recently an expansion of ŁΠ1/2 logic with fixed points has been considered [23]. In the present work we study the algebraic semantics of this logic, namely μŁΠ algebras, from algebraic, model theoretic and computational standpoints. We provide a characterisation of free μŁΠ algebras as a family of particular functions from [0,1]n to [0,1]. We show that the first-order theory of linearly ordered μŁΠ algebras enjoys quantifier elimination, being, more precisely, the model completion of the theory of linearly ordered ŁΠ1/2 algebras. Furthermore, we give a functional representation of any ŁΠ1/2 algebra in the style of Di Nola Theorem for MV-algebras and finally we prove that the equational theory of μŁΠ algebras is in PSPACE. © The Author 2010. Published by Oxford University Press. All rights reserved.Marchioni acknowledges partial support of the Spanish projects MULOG2 (TIN2007-68005-C04), Agreement Technologies (CONSOLIDER CSD2007-0022, INGENIO 2010), the Generalitat de Catalunya grant 2009-SGR-1434, and Juan de la Cierva Program of the Spanish MICINN, as well as the ESF Eurocores-LogICCC/MICINN project (FFI2008-03126-E/FILO). Spada acknowledges partially supported of the FWF project P 19872-N18.Peer Reviewe
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
Generic substitutions
Up to equivalence, a substitution in propositional logic is an endomorphism
of its free algebra. On the dual space, this results in a continuous function,
and whenever the space carries a natural measure one may ask about the
stochastic properties of the action. In classical logic there is a strong
dichotomy: while over finitely many propositional variables everything is
trivial, the study of the continuous transformations of the Cantor space is the
subject of an extensive literature, and is far from being a completed task. In
many-valued logic this dichotomy disappears: already in the finite-variable
case many interesting phenomena occur, and the present paper aims at displaying
some of these.Comment: 22 pages, 2 figures. Revised version according to the referee's
suggestions. To appear in the J. of Symbolic Logi