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On the representation of lattices by modules


Author: George Hutchinson
Journal: Trans. Amer. Math. Soc. 209 (1975), 311-351
MSC: Primary 06A20
DOI: https://doi.org/10.1090/S0002-9947-1975-0376462-5
MathSciNet review: 0376462
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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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DOI: https://doi.org/10.1090/S0002-9947-1975-0376462-5
Article copyright: © Copyright 1975 American Mathematical Society

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