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Unpublished Manuscript ◽  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 Math.37(1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving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.

scholarly journals Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices

2016 ◽  
Vol 45 (3/4) ◽  
Author(s):  
Marcin Łazarz
Keyword(s):  
Modular Lattice ◽  
Continuous Lattice ◽  
Atomic Lattice ◽  
Discrete Lattices ◽  
Upper Continuous Lattice ◽  
Upper Continuous ◽  
Strongly Atomic Lattice ◽  
Atomic Lattices

In the paper we investigate Birkhoff’s conditions (Bi) and (Bi*). We prove that a discrete lattice L satisfies the condition (Bi) (the condition (Bi*)) if and only if L is a 4-cell lattice not containing a cover-preserving sublattice isomorphic to the lattice S*7 (the lattice S7). As a corollary we obtain a well known result of J. Jakub´ık from [6]. Furthermore, lattices S7 and S*7 are considered as so-called partially cover-preserving sublattices of a given lattice L, S7 ≪ L and S7 ≪ L, in symbols. It is shown that an upper continuous lattice L satisfies (Bi*) if and only if L is a 4-cell lattice such that S7 ≪/ L. The final corollary is a generalization of Jakubík’s theorem for upper continuous and strongly atomic lattices. Keywords: Birkhoff’s conditions, semimodularity conditions, modular lattice, discrete lattices, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice, cell, 4-cell lattice.  

SOME ASPECTS OF MODULAR LATTICES WITH DUAL KRULL DIMENSION

2005 ◽  
Vol 04 (03) ◽  
pp. 237-244
Author(s):  
MARK L. TEPLY ◽  
SEOG HOON RIM
Keyword(s):  
Endomorphism Ring ◽  
Modular Lattice ◽  
Type Theorem ◽  
Krull Dimension ◽  
Local Domain ◽  
Module Theory ◽  
Modular Lattices ◽  
Atomic Lattices

For an ordinal α, a modular lattice L with 0 and 1 is α-atomic if L has dual Krull dimension α but each interval [0,x] with x < 1 has dual Krull dimension <α. The properties of α-atomic lattices are presented and applied to module theory. The endomorphism ring of certain types of α-atomic modules is a local domain and hence there is a Krull–Schmidt type theorem for those α-atomic modules.

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.

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.

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.

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.

scholarly journals Medians are Below Joins in Semimodular Lattices of Breadth 2

Order ◽  
2020 ◽  
Author(s):  
Gábor Czédli ◽  
Robert C. Powers ◽  
Jeremy M. White
Keyword(s):  
Finite Length ◽  
Path Length ◽  
Semimodular Lattice ◽  
Minimum Path ◽  
Covering Graph ◽  
Semimodular Lattices ◽  
Upper Semimodular Lattice

AbstractLet L be a lattice of finite length and let d denote the minimum path length metric on the covering graph of L. For any $\xi =(x_{1},\dots ,x_{k})\in L^{k}$ ξ = ( x 1 , … , x k ) ∈ L k , an element y belonging to L is called a median of ξ if the sum d(y,x1) + ⋯ + d(y,xk) is minimal. The lattice L satisfies the c1-median property if, for any $\xi =(x_{1},\dots ,x_{k})\in L^{k}$ ξ = ( x 1 , … , x k ) ∈ L k and for any median y of ξ, $y\leq x_{1}\vee \dots \vee x_{k}$ y ≤ x 1 ∨ ⋯ ∨ x k . Our main theorem asserts that if L is an upper semimodular lattice of finite length and the breadth of L is less than or equal to 2, then L satisfies the c1-median property. Also, we give a construction that yields semimodular lattices, and we use a particular case of this construction to prove that our theorem is sharp in the sense that 2 cannot be replaced by 3.

On the Word Problem for Orthocomplemented Modular Lattices

1989 ◽  
Vol 41 (6) ◽  
pp. 961-1004 ◽  
Author(s):  
Michael S. Roddy
Keyword(s):  
Word Problem ◽  
Division Ring ◽  
Modular Lattice ◽  
Modular Lattices ◽  
Commutative Field ◽  
Free Modular Lattice ◽  
Finitely Presented ◽  
Unsolvable Word Problem

In [16] Freese showed that the word problem for the free modular lattice on 5 generators is unsolvable. His proof makes essential use of Mclntyre's construction of a finitely presented field with unsolvable word problem [30]. (We follow Cohn [7] in calling what is commonly called a division ring a field, and what is commonly called a field a commutative field.) In this paper we will use similar ideas to obtain unsolvability results for varieties of modular ortholattices. The material in this paper is fairly wide ranging, the following are recommended as reference texts.

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