283 research outputs found
The variety generated by all the ordinal sums of perfect MV-chains
We present the logic BL_Chang, an axiomatic extension of BL (see P. H\'ajek -
Metamathematics of fuzzy logic - 1998, Kluwer) whose corresponding algebras
form the smallest variety containing all the ordinal sums of perfect MV-chains.
We will analyze this logic and the corresponding algebraic semantics in the
propositional and in the first-order case. As we will see, moreover, the
variety of BL_Chang-algebras will be strictly connected to the one generated by
Chang's MV-algebra (that is, the variety generated by all the perfect
MV-algebras): we will also give some new results concerning these last
structures and their logic.Comment: This is a revised version of the previous paper: the modifications
concern essentially the presentation. The scientific content is substantially
unchanged. The major variations are: Definition 2.7 has been improved.
Section 3.1 has been made more compact. A new reference, [Bus04], has been
added. There is some minor modification in Section 3.
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
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
Fuzzy Sets and Formal Logics
The paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics. Our focus is on the historical side of these developments. © 2015 Elsevier B.V. All rights reserved.partial support by the Spanish projects EdeTRI (TIN2012-39348- C02-01) and 2014 SGR 118.Peer reviewe
The Fuzzy Supersphere
We introduce the fuzzy supersphere as sequence of finite-dimensional,
noncommutative -graded algebras tending in a suitable limit to a dense
subalgebra of the -graded algebra of -functions on
the -dimensional supersphere. Noncommutative analogues of the body map
(to the (fuzzy) sphere) and the super-deRham complex are introduced. In
particular we reproduce the equality of the super-deRham cohomology of the
supersphere and the ordinary deRham cohomology of its body on the "fuzzy
level".Comment: 33 pages, LaTeX, some typos correcte
Twisted submanifolds of R^n
We propose a general procedure to construct noncommutative deformations of an
embedded submanifold of determined by a set of smooth
equations . We use the framework of Drinfel'd twist deformation of
differential geometry of [Aschieri et al., Class. Quantum Gravity 23 (2006),
1883]; the commutative pointwise product is replaced by a (generally
noncommutative) -product determined by a Drinfel'd twist. The twists we
employ are based on the Lie algebra of vector fields that are tangent
to all the submanifolds that are level sets of the ; the twisted Cartan
calculus is automatically equivariant under twisted tangent infinitesimal
diffeomorphisms. We can consistently project a connection from the twisted
to the twisted if the twist is based on a suitable Lie
subalgebra . If we endow with a metric
then twisting and projecting to the normal and tangent vector fields commute,
and we can project the Levi-Civita connection consistently to the twisted ,
provided the twist is based on the Lie subalgebra
of the Killing vector fields of the metric; a
twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can
be characterized in terms of generators and polynomial relations. We present in
some detail twisted cylinders embedded in twisted Euclidean and
twisted hyperboloids embedded in twisted Minkowski [these are
twisted (anti-)de Sitter spaces ].Comment: Latex file, 48 pages, 1 figure. Slightly adapted version to the new
preprint arXiv:2005.03509, where the present framework is specialized to
quadrics and other algebraic submanifolds of R^n. Several typos correcte
Dolan-Grady Relations and Noncommutative Quasi-Exactly Solvable Systems
We investigate a U(1) gauge invariant quantum mechanical system on a 2D
noncommutative space with coordinates generating a generalized deformed
oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge
covariant derivatives obeying the nonlinear Dolan-Grady relations. This
restricts the structure function of the deformed oscillator algebra to a
quadratic polynomial. The cases when the coordinates form the su(2) and sl(2,R)
algebras are investigated in detail. Reducing the Hamiltonian to 1D
finite-difference quasi-exactly solvable operators, we demonstrate partial
algebraization of the spectrum of the corresponding systems on the fuzzy sphere
and noncommutative hyperbolic plane. A completely covariant method based on the
notion of intrinsic algebra is proposed to deal with the spectral problem of
such systems.Comment: 25 pages; ref added; to appear in J. Phys.
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