Hjelmslev Planes Derived from Modular Lattices

1969 ◽  
Vol 21 ◽  
pp. 76-83 ◽  
Author(s):  
Benno Artmann
Keyword(s):  
Modular Lattice ◽  
Hjelmslev Plane ◽  
Modular Lattices ◽  
Natural Embedding ◽  
Valuation Rings ◽  
A Chain ◽  
Homogeneous Basis

In several papers, W. Klingenberg has elaborated the connections between Hjelmslev planes and a class of rings, called H-rings (4; 5; 6), which are rings of coordinates for the corresponding Hjelmslev planes. Certain homomorphic images of valuation rings are examples of H-rings. In these examples, the lattice of (right) ideals of the ring, say R,is a chain, and the coordinatization of the corresponding Hjelmslev plane yields a natural embedding of the plane in the lattice L(R3) of (right) submodules of the module R3. Now, L(R3) is a modular lattice with a homogeneous basis of order 3 given by the submodules a1 = (1, 0, 0)R, a2 = (0, 1, 0)R, a2 = (0, 0, 1)R, and the sublattices L(N, ai) of elements less than or equal to ai are chains. Forgetting about the ring, we find ourselves in the situation of a problem suggested by Skornyakov (7, Problem 23, p. 166), namely, to study modular lattices with a homogeneous basis of chains. Baer (2) and Inaba (3) investigated lattices of this kind with Desarguesian properties and assuming that the chains L(N, ai) were finite. Representations of the lattices by means of certain rings can be found in both articles.

Some Examples of Complemented Modular Lattices

1962 ◽  
Vol 5 (2) ◽  
pp. 111-121 ◽  
Author(s):  
G. Grätzer ◽  
Maria J. Wonenburger
Keyword(s):  
Boolean Algebra ◽  
Modular Lattice ◽  
Complete Boolean Algebra ◽  
Modular Lattices ◽  
Additive Property ◽  
Von Neumann ◽  
Continuous Geometry ◽  
Definition Of ◽  
Homogeneous Basis

Let L be a complemented, χ-complete modular lattice. A theorem of Amemiya and Halperin (see [l], Theorem 4.3) asserts that if the intervals [O, a] and [O, b], a, bεL, are upper χ-continuous then [O, a∪b] is also upper χ-continuous. Roughly speaking, in L upper χ-continuity is additive. The following question arises naturally: is χ-completeness an additive property of complemented modular lattices? It follows from Corollary 1 to Theorem 1 below that the answer to this question is in the negative.A complemented modular lattice is called a Von Neumann geometry or continuous geometry if it is complete and continuous. In particular a complete Boolean algebra is a Von Neumann geometry. In any case in a Von Neumann geometry the set of elements which possess a unique complement form a complete Boolean algebra. This Boolean algebra is called the centre of the Von Neumann geometry. Theorem 2 shows that any complete Boolean algebra can be the centre of a Von Neumann geometry with a homogeneous basis of order n (see [3] Part II, definition 3.2 for the definition of a homogeneous basis), n being any fixed natural integer.

On the notion of ranked complementedness in a lattice

2020 ◽  
pp. 2250004
Author(s):  
Francois Koch van Niekerk
Keyword(s):  
Natural Number ◽  
Modular Lattice ◽  
Finite Height ◽  
Number Formula ◽  
A Chain ◽  
Complemented Lattice

Not every element in a lattice has a complement. In this paper we introduce a notion of ranked complement, which depends on a natural number [Formula: see text], so that for every element [Formula: see text] in a lattice with finite height there exists [Formula: see text] such that [Formula: see text] has a complement of rank [Formula: see text]. One of the main results we establish is that in a modular lattice having finite height, every element has a complement of rank less than [Formula: see text] if and only if there is a chain [Formula: see text] of elements such that each interval [Formula: see text] is a complemented lattice.

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.

Chain Conditions for Modular Lattices with Finite Group Actions

1979 ◽  
Vol 31 (3) ◽  
pp. 558-564 ◽  
Author(s):  
Joe W. Fisher
Keyword(s):  
Modular Lattice ◽  
Chain Condition ◽  
Common Fixed Points ◽  
Finite Subgroup ◽  
Modular Lattices ◽  
Finite Group Actions ◽  
Chain Conditions ◽  
Combinatorial Result ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

scholarly journals ON COMPLETE CONGRUENCE LATTICES OF COMPLETE MODULAR LATTICES

1991 ◽  
Vol 01 (02) ◽  
pp. 147-160 ◽  
Author(s):  
R. FREESE ◽  
G. GRÄTZE ◽  
E. T. SCHMIDT
Keyword(s):  
Complete Lattice ◽  
Modular Lattice ◽  
Modular Lattices ◽  
Congruence Lattices ◽  
Congruence Relations

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In 1988, the second author announced the converse: every complete lattice L can be represented as the lattice of complete congruence relations of some complete lattice K. In this paper we improve this result by showing that K can be chosen to be a complete modular lattice.

Inertial Isomorphisms of V-Rings

1967 ◽  
Vol 19 ◽  
pp. 529-539 ◽  
Author(s):  
N. Heerema
Keyword(s):  
Power Series ◽  
Residue Field ◽  
Identity Mapping ◽  
Power Series Ring ◽  
Natural Embedding ◽  
Series Ring ◽  
Valuation Rings ◽  
Ramification Index ◽  
Rank One ◽  
The Common

Throughout this paper R and Rn will denote v-rings, that is, complete discrete rank-one valuation rings of characteristic zero, having a common residue field k of characteristic p. R is assumed unramified and Rn has ramification index n. Let π be a prime element in Rn. Then Rn = R[π], where π is a root of an Eisenstein polynomial ƒ = xn + pƒn-1 xn-1 + … + pƒ0 with coefficients in R and ƒ0 a unit. Thus Rn is inertially isomorphic to R[[x]]/ƒR[(x)], that is, the rings are isomorphic by a mapping which induces the identity mapping on the common residue field. R[[x]] represents the power series ring in the indeterminate x over R. In this paper we identify Rn with R[[x]]/ƒR[[x]], R with its natural embedding in Rn and π with x + ƒR[[x]].

scholarly journals Commutative rings with comparable regular elements

1996 ◽  
Vol 60 (1) ◽  
pp. 90-108 ◽  
Author(s):  
Paolo Zanardo
Keyword(s):  
Valuation Ring ◽  
Commutative Rings ◽  
The Other ◽  
Regular Elements ◽  
Zero Divisors ◽  
Valuation Rings ◽  
A Chain ◽  
Prüfer Rings ◽  
The Ideal

AbstractLet ℜ be the class of commutative rings R with comparable regular elements, that is, given two non zero-divisors in R, one divides the other. Applying the notion of V-valuation due to Harrison and Vitulli, we define the class V-val of V-valuated rings, which is contained in ℜ and contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜ but not in V-val; we prove that a ring R is a Prüfer valuation ring if and only if it is Prüfer and V-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the ideal I∞ of a ring R ∈ ℜ, which corresponds to the counterimage of ∞, whenever R is V-valuated.

Embedding Right Chain Rings in Chain Rings

1978 ◽  
Vol 30 (5) ◽  
pp. 1079-1086 ◽  
Author(s):  
H. H. Brungs ◽  
G. Törner
Keyword(s):  
Maximal Ideal ◽  
Coordinate Ring ◽  
Hjelmslev Plane ◽  
Chain Ring ◽  
Projective Hjelmslev Plane ◽  
Algebraic Version ◽  
Zero Divisors ◽  
Starting Point ◽  
Affine Hjelmslev Plane ◽  
A Chain

The following problem was the starting point for this investigation: Can every desarguesian affine Hjelmslev plane be embedded into a desarguesian projective Hjelmslev plane [8]? An affine Hjelmslev plane is called desarguesian if it can be coordinatized by a right chain ring R with a maximal ideal J(R) consisting of two-sided zero divisors. A projective Hjemslev plane is called desarguesian if the coordinate ring is in addition a left chain ring, i.e. a chain ring. This leads to the algebraic version of the above problem, namely the embedding of right chain rings into suitable chain rings. We can prove the following result.

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.

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.

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