The free modular lattice on four generators is not finitely presentable

1972 ◽  
Vol 2 (1) ◽  
pp. 284-285 ◽  
Author(s):  
T. Evans ◽  
D. Y. Hong
Keyword(s):  
Modular Lattice ◽  
Free Modular 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.

scholarly journals Defining relations of a free modular lattice of rank 3

2013 ◽  
Vol 57 (10) ◽  
pp. 59-61 ◽  
Author(s):  
A. G. Gein ◽  
M. P. Shushpanov
Keyword(s):  
Modular Lattice ◽  
Defining Relations ◽  
Free Modular Lattice

scholarly journals A note on distributive sublattices of a modular lattice

1969 ◽  
Vol 65 (2) ◽  
pp. 219-222 ◽  
Author(s):  
Raymond Balbes
Keyword(s):  
Modular Lattice

scholarly journals A Semimodular Imbedding of Lattices

1960 ◽  
Vol 12 ◽  
pp. 582-591 ◽  
Author(s):  
D. T. Finkbeiner
Keyword(s):  
Modular Lattice ◽  
Dimensional Lattice ◽  
General Lattice ◽  
Basic Properties ◽  
Arithmetic Properties ◽  
Finite Dimensional ◽  
Unpublished Work ◽  
Restricted Type ◽  
Point Lattice

The study of structural or arithmetic properties of a general lattice often can be facilitated by imbedding as a sublattice of a lattice of a more restricted type whose properties are known. However, if is too restricted, a general imbedding is impossible; for example, cannot be modular because , as a sublattice of , would then have to be modular. One of the best results of this nature has been given by Dilworth in an unpublished work in which he shows that any finite dimensional lattice is isomorphic to a sublattice of a semi-modular point lattice (1, pp. 105 and 110). In the present paper Dilworth's imbedding process is modified to obtain a sharper result: Any finite dimensional lattice is isometrically isomorphic to a sublattice of a semi-modular lattice which has the same number of points as and which preserves basic properties of the join-irreducible arithmetic of .

scholarly journals 2- and 3-Modular lattice wiretap codes in small dimensions

2015 ◽  
Vol 26 (6) ◽  
pp. 571-590 ◽  
Author(s):  
Fuchun Lin ◽  
Frédérique Oggier ◽  
Patrick Solé
Keyword(s):  
Modular Lattice ◽  
Wiretap Codes

scholarly journals CRITICAL POINTS OF PAIRS OF VARIETIES OF ALGEBRAS

2009 ◽  
Vol 19 (01) ◽  
pp. 1-40 ◽  
Author(s):  
PIERRE GILLIBERT
Keyword(s):  
Natural Number ◽  
Critical Points ◽  
Modular Lattice ◽  
General Theorem ◽  
Finitely Generated ◽  
Congruence Distributive

For a class [Formula: see text] of algebras, denote by Conc[Formula: see text] the class of all (∨, 0)-semilattices isomorphic to the semilattice ConcA of all compact congruences of A, for some A in [Formula: see text]. For classes [Formula: see text] and [Formula: see text] of algebras, we denote by [Formula: see text] the smallest cardinality of a (∨, 0)-semilattices in Conc[Formula: see text] which is not in Conc[Formula: see text] if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties [Formula: see text] and [Formula: see text], [Formula: see text] is either finite, or ℵn for some natural number n, or ∞. We also find two finitely generated modular lattice varieties [Formula: see text] and [Formula: see text] such that [Formula: see text], thus answering a question by J. Tůma and F. Wehrung.

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