An implication basis for linear forms |
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Authors: | R Padmanabhan P Penner |
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Institution: | (1) Department of Mathematics, University of Manitoba, Winnipeg, Man. R3T 2N2, Canada |
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Abstract: | The main results of this paper are a generalization of the results of S. Fajtlowicz and J. Mycielski on convex linear forms.
We show that if Vn is the variety generated by all possible algebras
, where R denotes the real numbers and
, for some
, then any basis for the set of all identities satisfied by Vn is infinite. But on the other hand, the identities satisfied by Vn are a consequence of gL and μn, where μn is the n-ary medial law and the inference rule gL is an implication patterned after the classical rigidity lemma of algebraic geometry. We also prove that the identities satisfied
by
are a consequence of gL and μn iff {p1, ... , pn} is algebraically independent. We then prove analagous results for algebras
of arbitrary type τ and in the final section of this paper, we show that analagous results hold for Abelian group hyperidentities.
This paper is dedicated to Walter Taylor.
Received July 16, 2005; accepted in final form January 12, 2006.
The research of both authors was supported by an operating grant ODP0008215 from NSERC. |
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Keywords: | 08B05 20N05 |
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