Non-Axiomatizability of the amalgamation class of modular lattice varieties

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 One of the best results of this nature has been given by Dilworth in an unpublished work in which he shows that any finite dimensional lattice is isomorphic to a sublattice of a semi-modular point lattice (1, pp. 105 and 110).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 .

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

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-015-0267-2 ◽

2015 ◽

Vol 26 (6) ◽

pp. 571-590 ◽

Cited By ~ 6

Author(s):

Fuchun Lin ◽

Frédérique Oggier ◽

Patrick Solé

Keyword(s):

Modular Lattice ◽

Wiretap Codes

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CRITICAL POINTS OF PAIRS OF VARIETIES OF ALGEBRAS

International Journal of Algebra and Computation ◽

10.1142/s0218196709004932 ◽

2009 ◽

Vol 19 (01) ◽

pp. 1-40 ◽

Cited By ~ 11

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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