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.  

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Unpublished Manuscript ◽  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 Math.37(1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving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

2009 ◽  
Vol 2 (1) ◽  
pp. 87-90
Author(s):  
R. M. H. Rahman
Keyword(s):  
Distributive Lattice ◽  
Partial Ordering ◽  
Continuous Lattice ◽  
Finite Distributive Lattice ◽  
Dense Lattice ◽  
Pseudocomplemented Lattice ◽  
Upper Continuous Lattice ◽  
Upper Continuous ◽  
Stone Lattice

By a new partial ordering relation "≤" the set of convex sublattices CS(L)  of a lattice L is again a lattice. In this paper we establish some results on the pseudocomplementation of (CS(L); ≤). We show that a lattice L with 0 is dense if and only if CS(L) is dense. Then we prove that a finite distributive lattice is a Stone lattice if and only if CS(L) is Stone. We also prove that an upper continuous lattice L is a Stone lattice if and only if CS(L) is Stone.  Keywords: Upper continuous lattice; Pseudocomplemented lattice; Dense lattice; Stone lattice. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.  DOI: 10.3329/jsr.v2i1.2485                  J. Sci. Res. 2 (1), 87-90  (2010) 

Lattices of Radicals of Ω-Groups

2006 ◽  
Vol 13 (03) ◽  
pp. 381-404
Author(s):  
G. L. Booth ◽  
Q. N. Petersen ◽  
S. Veldsman
Keyword(s):  
Complete Lattice ◽  
Modular Lattice ◽  
Universal Class ◽  
Atomic Lattice ◽  
Universal Classes ◽  
Definition Of ◽  
Associative Rings

Snider initiated the study of lattices of the class of radicals, in the sense of Kurosh and Amitsur, of associative rings. Various authors continued the investigation in more general universal classes. Recently, Fernández-Alonso et al. studied the lattice of all preradicals in R-Mod. Our definition of a preradical is weaker than theirs. In this paper, we consider the lattices of ideal maps 𝕀, preradical maps ℙ, Hoehnke radical maps ℍ and Plotkin radical maps 𝔹 in any universal class of Ω-groups (of the same type). We show that 𝕀 is a complete and modular lattice which contains atoms. In general, 𝕀 is not atomic. 𝕀 contains ℙ as a complete and atomic sublattice, whereas ℍ and 𝔹 are not sublattices of 𝕀. In its own right, ℍ is a complete and atomic lattice and 𝔹 is a complete lattice. We identify subclasses of 𝕀, ℙ and ℍ that are sublattices or preserve the meet (or join) of these respective lattices.

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.

scholarly journals Distributivity for Upper Continuous and Strongly Atomic Lattices

Studia Logica ◽  
2016 ◽  
Vol 105 (3) ◽  
pp. 471-478
Author(s):  
Marcin Łazarz ◽  
Krzysztof Siemieńczuk
Keyword(s):  
Upper Continuous ◽  
Atomic Lattices

The Duality of Distributive Continuous Lattices

1980 ◽  
Vol 32 (2) ◽  
pp. 385-394 ◽  
Author(s):  
B. Banaschewski
Keyword(s):  
Dual Space ◽  
Boolean Algebras ◽  
Distributive Lattices ◽  
Continuous Lattice ◽  
Prime Spectrum ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Lattices ◽  
Prime Elements

Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.

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.

scholarly journals Modularity for upper continuous and strongly atomic lattices

2016 ◽  
Vol 76 (4) ◽  
pp. 493-495 ◽  
Author(s):  
Marcin Łazarz ◽  
Krzysztof Siemieńczuk
Keyword(s):  
Upper Continuous ◽  
Atomic Lattices

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.

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