Dimension-complemented and locally finite dimensional cylindric algebras are elementarily equivalent

1981 ◽  
Vol 13 (1) ◽  
pp. 157-163 ◽  
Author(s):  
H. Andréka ◽  
I. Németi
Keyword(s):  
Cylindric Algebras ◽  
Locally Finite ◽  
Finite Dimensional

scholarly journals GRADED TWISTED CALABI–YAU ALGEBRAS ARE GENERALIZED ARTIN–SCHELTER REGULAR

2021 ◽  
pp. 1-54
Author(s):  
MANUEL L. REYES ◽  
DANIEL ROGALSKI
Keyword(s):  
Regularity Property ◽  
Graded Algebra ◽  
General Study ◽  
Locally Finite ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Dimension 2 ◽  
Separable Algebras ◽  
Gk Dimension

Abstract This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

scholarly journals How low can vacuum energy go when your fields are finite-dimensional?

2019 ◽  
Vol 28 (14) ◽  
pp. 1944006
Author(s):  
ChunJun Cao ◽  
Aidan Chatwin-Davies ◽  
Ashmeet Singh
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Degrees Of Freedom ◽  
Vacuum Energy ◽  
Quantum Field ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Klein Gordon ◽  
Dimensional Version

According to the holographic bound, there is only a finite density of degrees of freedom in space when gravity is taken into account. Conventional quantum field theory does not conform to this bound, since in this framework, infinitely many degrees of freedom may be localized to any given region of space. In this paper, we explore the viewpoint that quantum field theory may emerge from an underlying theory that is locally finite-dimensional, and we construct a locally finite-dimensional version of a Klein–Gordon scalar field using generalized Clifford algebras. Demanding that the finite-dimensional field operators obey a suitable version of the canonical commutation relations makes this construction essentially unique. We then find that enforcing local finite dimensionality in a holographically consistent way leads to a huge suppression of the quantum contribution to vacuum energy, to the point that the theoretical prediction becomes plausibly consistent with observations.

Nonfinitizability of classes of representable cylindric algebras

1969 ◽  
Vol 34 (3) ◽  
pp. 331-343 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Predicate Logic ◽  
Infinite Dimension ◽  
Fundamental Result ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Locally Finite ◽  
First Order ◽  
Alfred Tarski ◽  
Sentential Logic ◽  
Algebraic Counterpart

Cylindric algebras were introduced by Alfred Tarski about 1952 to provide an algebraic analysis of (first-order) predicate logic. With each cylindric algebra one can, in fact, associate a certain, in general infinitary, predicate logic; for locally finite cylindric algebras of infinite dimension the associated predicate logics are finitary. As with Boolean algebras and sentential logic, the algebraic counterpart of completeness is representability. Tarski proved the fundamental result that every locally finite cylindric algebra of infinite dimension is representable.

STONE SPACE OF CYLINDRIC ALGEBRAS AND TOPOLOGICAL MODEL SPACES

2016 ◽  
Vol 81 (3) ◽  
pp. 1069-1086
Author(s):  
CHARLES C. PINTER
Keyword(s):  
Boolean Algebra ◽  
Model Theory ◽  
Representation Theorem ◽  
Stone Space ◽  
Equivalence Relations ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Locally Finite ◽  
Abstract Model Theory ◽  
Image Position

AbstractThe Stone representation theorem was a milestone for the understanding of Boolean algebras. From Stone’s theorem, every Boolean algebra is representable as a field of sets with a topological structure. By means of this, the structural elements of any Boolean algebra, as well as the relations between them, are represented geometrically and can be clearly visualized. It is no different for cylindric algebras: Suppose that ${\frak A}$ is a cylindric algebra and ${\cal S}$ is the Stone space of its Boolean part. (Among the elements of the Boolean part are the diagonal elements.) It is known that with nothing more than a family of equivalence relations on ${\cal S}$ to represent quantifiers, ${\cal S}$ represents the full cylindric structure just as the Stone space alone represents the Boolean structure. ${\cal S}$ with this structure is called a cylindric space.Many assertions about cylindric algebras can be stated in terms of elementary topological properties of ${\cal S}$. Moreover, points of ${\cal S}$ may be construed as models, and on that construal ${\cal S}$ is called a model space. Certain relations between points on this space turn out to be morphisms between models, and the space of models with these relations hints at the possibility of an “abstract” model theory. With these ideas, a point-set version of model theory is proposed, in the spirit of pointless topology or category theory, in which the central insight is to treat the semantic objects (models) homologously with the corresponding syntactic objects so they reside together in the same space.It is shown that there is a new, purely algebraic way of introducing constants in cylindric algebras, leading to a simplified proof of the representation theorem for locally finite cylindric algebras. Simple rich algebras emerge as homomorphic images of cylindric algebras. The topological version of this theorem is especially interesting: The Stone space of every locally finite cylindric algebra ${\frak A}$ can be partitioned into subspaces which are the Stone spaces of all the simple rich homomorphic images of ${\frak A}$. Each of these images completely determines a model of ${\frak A}$, and all denumerable models of ${\frak A}$ appear in this representation.The Stone space ${\cal S}$ of every cylindric algebra can likewise be partitioned into closed sets which are duals of all the types in ${\frak A}$. This fact yields new insights into miscellaneous results in the model theory of saturated models.

scholarly journals Sums of finite-dimensional spaces

1969 ◽  
Vol 1 (3) ◽  
pp. 357-361 ◽  
Author(s):  
B.R. Wenner
Keyword(s):  
Metric Spaces ◽  
Dimension Theory ◽  
General Hypothesis ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Locally Countable

Analogues are developed to the sum theorems in the dimension theory of metric spaces. It is shown that, within the class of metric spaces, any locally countable, σ-locally finite, or closure-preserving sum of finite-dimensional sets is countable-dimensional. Similar results are obtained under the more general hypothesis of countable-dimensional rather than finite-dimensional sets.

Semisimplicity of Locally Finite Graded R#H-Modules

2009 ◽  
Vol 16 (01) ◽  
pp. 109-122
Author(s):  
Thomas Guédénon
Keyword(s):  
Abelian Group ◽  
Smash Product ◽  
Module Algebra ◽  
Locally Finite ◽  
Finite Dimensional

Let k be a field, Γ an abelian group with a bicharacter, R a colour algebra over k (i.e., a Γ-graded associative k-algebra with identity), H a Hopf colour k-algebra acting on R in such a way that R is a graded H-module algebra and the associated smash product R#H is a colour algebra. The aim of this paper is to study the semisimplicity of the category of H-locally finite Γ-graded R#H-modules. From our main result we deduce that if H is finite-dimensional and R is left graded-noetherian and graded-semisimple, then the colour algebra R#H is graded-semisimple if either H is graded-semisimple or if H is colour-cocommutative and R is colour-commutative and projective in the category of graded R#H-modules.

scholarly journals The Hilbert space of quantum gravity is locally finite-dimensional

2017 ◽  
Vol 26 (12) ◽  
pp. 1743013 ◽  
Author(s):  
Ning Bao ◽  
Sean M. Carroll ◽  
Ashmeet Singh
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Quantum Gravity ◽  
Density Operator ◽  
Small Region ◽  
Quantum Field ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Model Independent

We argue in a model-independent way that the Hilbert space of quantum gravity is locally finite-dimensional. In other words, the density operator describing the state corresponding to a small region of space, when such a notion makes sense, is defined on a finite-dimensional factor of a larger Hilbert space. Because quantum gravity potentially describes superpositions of different geometries, it is crucial that we associate Hilbert-space factors with spatial regions only on individual decohered branches of the universal wave function. We discuss some implications of this claim, including the fact that quantum-field theory cannot be a fundamental description of nature.

A characterization of piecewise linear homology manifolds

1983 ◽  
Vol 93 (2) ◽  
pp. 271-274 ◽  
Author(s):  
W. J. R. Mitchell
Keyword(s):  
Piecewise Linear ◽  
Subsequent Paper ◽  
Locally Compact ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Set ◽  
Topological Manifolds ◽  
Homology Manifolds

We state and prove a theorem which characterizes piecewise linear homology manifolds of sufficiently large dimension among locally compact finite-dimensional absolute neighbourhood retracts (ANRs). The proof is inspired by Cannon's observation (3) that a piecewise linear homology manifold is a topological manifold away from a locally finite set, and uses Galewski and Stern's work on simplicial triangulations of topological manifolds, the Edwards–Cannon–Quinn characterization of topological manifolds and Siebenmann's work on ends (3, 6, 4, 13, 14, 15, 16). All these tools have suitable relative versions and so the theorem can be extended to the bounded case. However, the most satisfactory extension requires a classification of triangulations of homology manifolds up to concordance. This will be given in a subsequent paper and the bounded case will be postponed to that paper.

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