algebra universalis

, Volume 7, Issue 1, pp 341–356 | Cite as

On the existence of subalgebras of direct products with prescribedd-fold projections

  • George M. Bergman


Baker and Pixley have shown the equivalence of a number of conditions on a positive integerd and a varietyV be uniquely determined by its projections in thed-fold subproductsA i (1)×...×A i (d). It is shown here thatunder Baker and Pixley's conditions this uniqueness result is complemented by an existence result: SupposeA 1,...,A r V, and that for everyd-tupleI={i(1),...,i(d)} a subalgebraS I A i(1)×...×A i(d) is given. Then these data are the projections of one subalgebraS⊆ A 1×...A r if and only if they are “consistent” on eachd+1-tuple {i(1),...,i(d+1)}.

In the case where eachA i is the lattice {0, 1}, these results lead to the well-known description of finite distributive lattices in terms of finite partially ordered sets.

Under appropriate hypotheses the above result generalizes to subalgebras of infinite direct products.


Direct Product Variety Versus Subdirect Product Versus Satisfy Infinite Cardinal 
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  1. [1]
    K. A. Baker andA. F. Pixley,Polynomial interpolation and the Chinese remainder theorem for algebraic systems. Math. Zeitschr.143 (1975), 165–174MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag 1977

Authors and Affiliations

  • George M. Bergman
    • 1
  1. 1.University of California at BerkeleyBerkeleyUSA

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