On the representation of lattices by modules

Hutchinson, George

doi:10.1090/S0002-9947-1975-0376462-5

On the representation of lattices by modules
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by George Hutchinson

Trans. Amer. Math. Soc. 209 (1975), 311-351

DOI: https://doi.org/10.1090/S0002-9947-1975-0376462-5

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Abstract:

For a commutative ring $R$ with unit, a lattice $L$ is “representable by $R$-modules” if $L$ is embeddable in the lattice of submodules of some unitary left $R$-module. A procedure is given for generating an infinite first-order axiomatization of the class of all lattices representable by $R$-modules. Each axiom is a universal Horn formula for lattices. The procedure for generating the axioms is closely related to the ring structure, and is “effective” in the sense that many nontrivial axioms can be obtained by moderate amounts of computation.

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