6,936 research outputs found
Modal multilattice logics with Tarski, Kuratowski, and Halmos operators
In this paper, we consider modal multilattices with Tarski, Kuratowski, and Halmos closure and interior operators as well as the corresponding logics which are multilattice versions of the modal logics MNT4, S4, and S5, respectively. The former modal multilattice logic is a new one. The latter two modal multilattice logics have been already mentioned in the literature, but algebraic completeness results have not been established for them before. We present a multilattice version of MNT4 in a form of a sequent calculus and prove the algebraic and neighbourhood completeness theorems for it. We extend the algebraic completeness result for the multilattice versions of S4 and S5 as well
Algebraic Quantum Mechanics and Pregeometry
We discuss the relation between the q-number approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the q-numbers with the elements of an algebra and regarding the primitive idempotents as "generalized points" we suggest an approach that may make it possible to dispense with an a priori given space manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of "neighbourhood operators", which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra
Algebraic totality, towards completeness
Finiteness spaces constitute a categorical model of Linear Logic (LL) whose
objects can be seen as linearly topologised spaces, (a class of topological
vector spaces introduced by Lefschetz in 1942) and morphisms as continuous
linear maps. First, we recall definitions of finiteness spaces and describe
their basic properties deduced from the general theory of linearly topologised
spaces. Then we give an interpretation of LL based on linear algebra. Second,
thanks to separation properties, we can introduce an algebraic notion of
totality candidate in the framework of linearly topologised spaces: a totality
candidate is a closed affine subspace which does not contain 0. We show that
finiteness spaces with totality candidates constitute a model of classical LL.
Finally, we give a barycentric simply typed lambda-calculus, with booleans
and a conditional operator, which can be interpreted in this
model. We prove completeness at type for
every n by an algebraic method
Generalized Vietoris Bisimulations
We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our
notion of bisimulation is sound and complete for behavioural equivalence, and
generalizes Vietoris bisimulations [4]. The main result of our paper is that
bisimulation for a coalgebra is the topological closure of
bisimulation for the underlying coalgebra
Conditionals and modularity in general logics
In this work in progress, we discuss independence and interpolation and
related topics for classical, modal, and non-monotonic logics
Natural models of theories of green points
We explicitly present expansions of the complex field which are models of the
theories of green points in the multiplicative group case and in the case of an
elliptic curve without complex multiplication defined over . In
fact, in both cases we give families of structures depending on parameters and
prove that they are all models of the theories, provided certain instances of
Schanuel's conjecture or an analogous conjecture for the exponential map of the
elliptic curve hold. In the multiplicative group case, however, the results are
unconditional for generic choices of the parameters
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