7 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
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
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
DUALITIES AND REPRESENTATIONS FOR MANY-VALUED LOGICS IN THE HIERARCHY OF WEAK NILPOTENT MINIMUM.
In this thesis we study particular subclasses of WNM algebras.
The variety of WNM algebras forms the algebraic semantics of the
WNM logic, a propositional many-valued logic that generalizes some
well-known case in the setting of triangular norms logics.
WNM logic lies in the hierarchy of schematic extensions of MTL, which is
proven to be the logic of all left-continuous triangular norms and their residua.
In this work, I have extensively studied two extensions
of WNM logic, namely RDP logic and NMG logic, from the point of view of
algebraic and categorical logic.
We develop spectral dualities between the varieties of algebras
corresponding to RDP logic and NMG logic, and suitable defined combinatorial categories.
Categorical dualities allow to give algorithmic construction of products in
the dual categories obtaining computable descriptions of coproducts
(which are notoriously hard to compute working only in the algebraic side)
for the corresponding finite algebras. As a byproduct, representation theorems
for finite algebras and free finitely generated algebras in the considered varieties
are obtained. This latter characterization is especially useful to provide explicit
construction of a number of objects relevant from the point of view of the logical
interpretation of the varieties of algebras: normal forms, strongest deductive
interpolants and most general unifiers
Neutrality and Many-Valued Logics
In this book, we consider various many-valued logics: standard, linear,
hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We
survey also results which show the tree different proof-theoretic frameworks
for many-valued logics, e.g. frameworks of the following deductive calculi:
Hilbert's style, sequent, and hypersequent. We present a general way that
allows to construct systematically analytic calculi for a large family of
non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and
p-adic valued logics characterized by a special format of semantics with an
appropriate rejection of Archimedes' axiom. These logics are built as different
extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's,
Product, and Post's logics). The informal sense of Archimedes' axiom is that
anything can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and it
is not well-founded and well-ordered. On the base of non-Archimedean valued
logics, we construct non-Archimedean valued interval neutrosophic logic INL by
which we can describe neutrality phenomena.Comment: 119 page