scholarly journals Review of the book of Vladimir Kovalevsky "Geometry of Locally Finite Spaces"

2008 ◽  
Vol 9 (2) ◽  
pp. 311-313
Author(s):  
Sergei Matveev
Keyword(s):  
Locally Finite ◽  
Finite Spaces

scholarly journals Properties of space set topological spaces

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2475-2487 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Topological Space ◽  
Crucial Role ◽  
Topological Structure ◽  
Special Kind ◽  
Topological Spaces ◽  
Separation Axiom ◽  
Locally Finite ◽  
Alexandroff Space ◽  
Finite Spaces ◽  
Point Set

Since a locally finite topological structure plays an important role in the fields of pure and applied topology, the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, SST) and further, proves that an SST is an Alexandroff space satisfying the separation axiom T0. Unlike a point set topology, since each element of an SST is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space (X,T) with |X| = 2 the axioms T0, semi-T1/2 and T1/2 are proved to be equivalent to each other. Furthermore, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected SST can be a good example showing that the separation axiom semi-T1/2 does not imply T1/2.

Examples of theories of presheaf type

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Finite Groups ◽  
Linear Independence ◽  
Geometric Theory ◽  
Vector Spaces ◽  
Linear Orders ◽  
Locally Finite ◽  
Strong Unit ◽  
Locally Finite Groups ◽  
Simplicial Sets ◽  
Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

Existentially closed central extensions of locally finite p-groups

1986 ◽  
Vol 100 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Felix Leinen ◽  
Richard E. Phillips
Keyword(s):  
Central Extensions ◽  
Locally Finite ◽  
Existentially Closed ◽  
Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

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.

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