950 research outputs found
Lukasiewicz logic and Riesz spaces
We initiate a deep study of {\em Riesz MV-algebras} which are MV-algebras
endowed with a scalar multiplication with scalars from . Extending
Mundici's equivalence between MV-algebras and -groups, we prove that
Riesz MV-algebras are categorically equivalent with unit intervals in Riesz
spaces with strong unit. Moreover, the subclass of norm-complete Riesz
MV-algebras is equivalent with the class of commutative unital C-algebras.
The propositional calculus that has Riesz MV-algebras as
models is a conservative extension of \L ukasiewicz -valued
propositional calculus and it is complete with respect to evaluations in the
standard model . We prove a normal form theorem for this logic,
extending McNaughton theorem for \L ukasiewicz logic. We define the notions of
quasi-linear combination and quasi-linear span for formulas in and we relate them with the analogue of de Finetti's coherence
criterion for .Comment: To appear in Soft Computin
The Logic of Quasi-MV Algebras
The algebraic theory of quasi-MV algebras, generalizations of MV algebras arising in quantum computation, is by now rather well-developed. Although it is possible to define several interesting logics from these structures, so far this aspect has not been investigated. The present article aims at filling this ga
Towards understanding the Pierce-Birkhoff conjecture via MV-algebras
Our main issue was to understand the connection between \L ukasiewicz logic
with product and the Pierce-Birkhoff conjecture, and to express it in a
mathematical way. To do this we define the class of \textit{f}MV-algebras,
which are MV-algebras endowed with both an internal binary product and a scalar
product with scalars from . The proper quasi-variety generated by
, with both products interpreted as the real product, provides the
desired framework: the normal form theorem of its corresponding logical system
can be seen as a local version of the Pierce-Birkhoff conjecture
On some Properties of quasi-MV â Algebras and quasi-MV Algebras. Part IV
In the present paper, which is a sequel to [20, 4, 12], we investigate further the structure theory of quasi-MV algebras and â quasi-MV algebras. In particular: we provide a new representation of arbitrary â qMV algebras in terms of â qMV algebras arising out of their MV* term subreducts of regular elements; we investigate in greater detail the structure of the lattice of â qMV varieties, proving that it is uncountable, providing equational bases for some of its members, as well as analysing a number of slices of special interest; we show that the variety of â qMV algebras has the amalgamation property; we provide an axiomatisation of the 1-assertional logic of â qMV algebras; lastly, we reconsider the correspondence between Cartesian â qMV algebras and a category of Abelian lattice-ordered groups with operators first addressed in [10]
Representation of Perfect and Local MV-algebras
We describe representation theorems for local and perfect MV-algebras in
terms of ultraproducts involving the unit interval [0,1]. Furthermore, we give
a representation of local Abelian lattice-ordered groups with strong unit as
quasi-constant functions on an ultraproduct of the reals. All the above
theorems are proved to have a uniform version, depending only on the
cardinality of the algebra to be embedded, as well as a definable construction
in ZFC. The paper contains both known and new results and provides a complete
overview of representation theorems for such classes
Interval valued (\in,\ivq)-fuzzy filters of pseudo -algebras
We introduce the concept of quasi-coincidence of a fuzzy interval value with
an interval valued fuzzy set. By using this new idea, we introduce the notions
of interval valued (\in,\ivq)-fuzzy filters of pseudo -algebras and
investigate some of their related properties. Some characterization theorems of
these generalized interval valued fuzzy filters are derived. The relationship
among these generalized interval valued fuzzy filters of pseudo -algebras
is considered. Finally, we consider the concept of implication-based interval
valued fuzzy implicative filters of pseudo -algebras, in particular, the
implication operators in Lukasiewicz system of continuous-valued logic are
discussed
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