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Embedding and unsolvability theorems for modular lattices

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Abstract

LetR be a nontrivial ring with 1 and δ a cardinal. Let,L(R, δ) denote the lattice of submodules of a free unitaryR-module on δ generators. Let ℳ be the variety of modular lattices. A lattice isR-representable if embeddable in the lattice of submodules of someR-module; ℒ(R) denotes the quasivariety of allR-representable lattices. Let ω denote aleph-null, and let a (m, n) presentation havem generators andn relations,m, n≤ω.

THEOREM. There exists a (5, 1) modular lattice presentation having a recursively unsolvable word problem for any quasivarietyV,V ⊂ ℳ, such thatL(R, ω) is inV.

THEOREM. IfL is a denumerable sublattice ofL(R, δ), then it is embeddable in some sublatticeK ofL(R,δ*) having five generators, where δ*=δ for infinite δ and δ*=4δ(m+1) if δ is finite andL has a set ofm generators.

THEOREM. The free ℒ(R)-lattice on ω generators is embeddable in the free ℒ(R)-lattice on five generators.

THEOREM. IfL has an (m, n), ℒ(R)-presentation for denumerablem and finiten, thenL is embeddable in someK having a (5, 1) ℒ(R)-presentation.

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Authors and Affiliations

  1. National Institutes of Health, Bethesda, Maryland, USA

    George Hutchinson

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  1. George Hutchinson
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Hutchinson, G. Embedding and unsolvability theorems for modular lattices. Algebra Universalis 7, 47–84 (1977). https://doi.org/10.1007/BF02485417

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