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.

scholarly journals On congruence lattices of m-complete lattices

1992 ◽  
Vol 52 (1) ◽  
pp. 57-87 ◽  
Author(s):  
G. Grätzer ◽  
H. Lakser
Keyword(s):  
Complete Lattice ◽  
Extension Theorem ◽  
Unit Element ◽  
Algebraic Lattice ◽  
Converse Statement ◽  
Complete Lattices ◽  
Point Extension ◽  
Congruence Lattices ◽  
Congruence Relations ◽  
The One

AbstractThe lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In an earlier paper, we characterize this lattice as a complete lattice. Let m be an uncountable regular cardinal. The lattice L of all m-complete congruence relations of an m-complete lattice K is an m-algebraic lattice; if K is bounded, then the unit element of L is m-compact. Our main result is the converse statement: For an m-algebraic lattice L with an m-compact unit element, we construct a bounded m-complete lattice K such that L is isomorphic to the lattice of m-complete congruence relations of K. In addition, if L has more than one element, then we show how to construct K so that it will also have a prescribed automorphism group. On the way to the main result, we prove a technical theorem, the One Point Extension Theorem, which is also used to provide a new proof of the earlier result.

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.

Congruence lattices of complemented modular lattices

1984 ◽  
Vol 18 (3) ◽  
pp. 386-395 ◽  
Author(s):  
E. T. Schmidt
Keyword(s):  
Modular Lattices ◽  
Congruence Lattices

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.

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.

scholarly journals Congruence intersection properties for varieties of algebras

1999 ◽  
Vol 67 (1) ◽  
pp. 104-121 ◽  
Author(s):  
Paolo Agliano ◽  
Kirby A. Baker
Keyword(s):  
Principal Congruence ◽  
Intersection Property ◽  
Congruence Lattices ◽  
Congruence Relations ◽  
Compact Congruence ◽  
Proper Perspective ◽  
Congruence Distributive

AbstractIt is shown that a variety ν has distributive congruence lattices if and only if the intersection of two principal congruence relations is definable by equations involving terms with parameters. The nature of the terms involved then provides a useful classification of congruence distributive varieties. In particular, the classification puts into proper perspective two stronger properties. A variety is said to have the Principal Intersection Property if the intersection of any two principal congruence relations is principal, or the Compact Intersection Property if the intersection of two compact congruence relations is compact. For non-congruence-distributive varieties, it is shown that some useful constuctions are nevertheless possible.

scholarly journals L-Fuzzy Congruences and L-Fuzzy Kernel Ideals in Ockham Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Teferi Getachew Alemayehu ◽  
Derso Abeje Engidaw ◽  
Gezahagne Mulat Addis
Keyword(s):  
Complete Lattice ◽  
Ockham Algebra ◽  
Truth Values ◽  
Kernel Ideal ◽  
Congruence Relations ◽  
Fuzzy Ideal ◽  
Fuzzy Congruence ◽  
Distributive Law ◽  
Ockham Algebras ◽  
Equivalent Conditions

In this paper, we study fuzzy congruence relations and kernel fuzzy ideals of an Ockham algebra A , f , whose truth values are in a complete lattice satisfying the infinite meet distributive law. Some equivalent conditions are derived for a fuzzy ideal of an Ockham algebra A to become a fuzzy kernel ideal. We also obtain the smallest (respectively, the largest) fuzzy congruence on A having a given fuzzy ideal as its kernel.

Sign in / Sign up

Export Citation Format

Share Document