Department of Mathematics
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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 V_n is the variety generated by all algebras A=<R;f>, where R denotes the real numbers and f(x₁,...,x_n) = p₁x₁+...+p_nx_n, for some real numbers p₁,..., p_n, then any basis for the set of all identities satisfied by V_n is infinite. But on the other hand, the identities satisfied by V_n are a consequence of gL amd m, where m 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 A=<R;f> are a consequence of gL and m iff {p₁,...,p_n} is algebraically independent. We then prove analagous results for algebras A=<R;F> of arbitrary type and in the final section of this paper, we show that analogous results hold for Abelian group hyperidentities.