scholarly journals Meet representations in upper continuous modular lattices

1966 ◽  
Author(s):  
James Elton Delany
Keyword(s):  
Modular Lattices ◽  
Upper Continuous

scholarly journals GOLDIE DIMENSION, DUAL KRULL DIMENSION AND SUBDIRECT IRREDUCIBILITY

2010 ◽  
Vol 52 (A) ◽  
pp. 19-32 ◽  
Author(s):  
TOMA ALBU
Keyword(s):  
Krull Dimension ◽  
Subdirectly Irreducible ◽  
Modular Lattices ◽  
Survey Paper ◽  
Goldie Dimension ◽  
Subdirect Irreducibility ◽  
Upper Continuous ◽  
Torsion Theories ◽  
Quotient Modules ◽  
Noetherian Modules

AbstractIn this survey paper we present some results relating the Goldie dimension, dual Krull dimension and subdirect irreducibility in modules, torsion theories, Grothendieck categories and lattices. Our interest in studying this topic is rooted in a nice module theoretical result of Carl Faith [Commun. Algebra27 (1999), 1807–1810], characterizing Noetherian modules M by means of the finiteness of the Goldie dimension of all its quotient modules and the ACC on its subdirectly irreducible submodules. Thus, we extend his result in a dual Krull dimension setting and consider its dualization, not only in modules, but also in upper continuous modular lattices, with applications to torsion theories and Grothendieck categories.

An extension of F. Šik’s theorem on modular lattices

2018 ◽  
Vol 68 (6) ◽  
pp. 1321-1326
Author(s):  
Marcin Łazarz
Keyword(s):  
Finite Length ◽  
Modular Lattice ◽  
Unpublished Manuscript ◽  
Modular Lattices ◽  
Locally Finite ◽  
Semimodular Lattices ◽  
Upper Continuous ◽  
Atomic Lattices

AbstractJ. Jakubík noted in [JAKUBÍK, J.:Modular Lattice of Locally Finite Length, Acta Sci. Math.37(1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving sublattices isomorphic toS7. In the present paper we extend the scope of Šik’s theorem to the class of upper semimodular, upper continuous and strongly atomic lattices. Moreover, we show that corresponding result of Jakubík from [JAKUBÍK, J.:Modular Lattice of Locally Finite Length, Acta Sci. Math.37(1975), 79–82] cannot be strengthened is analogous way.

MODULAR QFD LATTICES WITH APPLICATIONS TO GROTHENDIECK CATEGORIES AND TORSION THEORIES

2004 ◽  
Vol 03 (04) ◽  
pp. 391-410 ◽  
Author(s):  
TOMA ALBU ◽  
MIHAI IOSIF ◽  
MARK L. TEPLY
Keyword(s):  
Modular Lattice ◽  
Independent Set ◽  
Torsion Theory ◽  
Modular Lattices ◽  
Finite Dimensional ◽  
Upper Continuous ◽  
Gabriel Dimension ◽  
Torsion Theories ◽  
Condition C ◽  
Compactly Generated

A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for any x∈L. We extend some results about QFD modules to upper continuous modular lattices by using Lemonnier's Lemma. One result says that QFD for a compactly generated lattice L is equivalent to Condition (C): for every m∈L, there exists a compact element t of L such that t∈[0,m] and [t,m[ has no maximal element. If L is not compactly generated, then QFD and (C) separate into two distinct conditions, which are analyzed and characterized for upper continuous modular lattices. We also extend to upper continuous modular lattices some characterizations of QFD modules with Gabriel dimension. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki theorem (I)

1996 ◽  
Vol 120 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Toma Albu ◽  
Patrick F. Smith
Keyword(s):  
Artinian Ring ◽  
Chain Condition ◽  
Torsion Theory ◽  
Krull Dimension ◽  
Main Body ◽  
Modular Lattices ◽  
Upper Continuous ◽  
Descending Chain Condition ◽  
Unital Rings

The Hopkins–Levitzki Theorem, discovered independently in 1939 by C. Hopkins and J. Levitzki states that a right Artinian ring with identity is right Noetherian. In the 1970s and 1980s it has been generalized to modules over non-unital rings by Shock[10], to modules satisfying the descending chain condition relative to a heriditary torsion theory by Miller-Teply[7], to Grothendieck categories by Năstăsescu [8], and to upper continuous modular lattices by Albu [1]. The importance of the relative Hopkins-Levitzki Theorem in investigating the structure of some relevant classes of modules, including injectives as well as projectives is revealed in [3] and [6], where the main body of both these monographs deals with this topic. A discussion on the various forms of the Hopkins–Levitzki Theorem for modules and Grothendieck categories and the connection between them may be found in [3].

Complemented Modular Lattices

1959 ◽  
Vol 11 ◽  
pp. 481-520 ◽  
Author(s):  
Ichiro Amemiya ◽  
Israel Halperin
Keyword(s):  
Modular Lattice ◽  
Modular Lattices ◽  
Von Neumann ◽  
Upper Continuous ◽  
Continuous Geometry

1.1 This paper gives a lattice theoretic investigation of “finiteness“ and “continuity of the lattice operations” in a complemented modular lattice. Although we usually assume that the lattice is-complete for some infinite,3we do not require completeness and continuity, as von Neumann does in his classical memoir on continuous geometry (3); nor do we assume orthocomplementation as Kaplansky does in his remarkable paper (1).1.2. Our exposition is elementary in the sense that it can be read without reference to the literature. Our brief preliminary § 2 should enable the reader to read this paper independently.1.3. Von Neumann's theory of independence (3, Part I, Chapter II) leans heavily on the assumption that the lattice is continuous, or at least upper continuous.

On sets of points of approximate continuity and ϱ-upper continuity

2020 ◽  
Vol 70 (2) ◽  
pp. 305-318
Author(s):  
Anna Kamińska ◽  
Katarzyna Nowakowska ◽  
Małgorzata Turowska
Keyword(s):  
Real Function ◽  
Approximate Continuity ◽  
Measurable Set ◽  
Upper Continuous ◽  
Lebesgue Measurable Set ◽  
Upper Continuity

Abstract In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

scholarly journals Seven-modular lattices and a septic base Jacobi identity

2003 ◽  
Vol 99 (2) ◽  
pp. 361-372 ◽  
Author(s):  
Heng Huat Chan ◽  
Kok Seng Chua ◽  
Patrick Solé
Keyword(s):  
Jacobi Identity ◽  
Modular Lattices

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

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