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An implication basis for linear forms. (English) Zbl 1108.08006
Summary: 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 $$\mathbf V_n$$ is the variety generated by all possible algebras $$\mathcal A = \left\langle \mathbf R;f \right\rangle$$, where $$\mathbf R$$ denotes the real numbers and $$f(x_1,\ldots,x_n) = p_1x_1 + \cdots + p_nx_n$$, for some $$p_1,\ldots,p_n \in \mathbf R$$, then any basis for the set of all identities satisfied by $$\mathbf V_n$$ is infinite. But on the other hand, the identities satisfied by $$\mathbf V_n$$ are a consequence of $$gL$$ and $$\mu_n$$, where $$\mu_n$$ 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 $$\mathcal A = \left\langle \mathbf R;f \right\rangle$$ are a consequence of $$gL$$ and $$\mu_{n}$$ iff $$\{p_1,\dots,p_n\}$$ is algebraically independent. We then prove analagous results for algebras $$\mathcal A = \left\langle \mathbf R;F \right\rangle$$ of arbitrary type $$\tau$$ and in the final section of this paper, we show that analagous results hold for Abelian group hyperidentities.

##### MSC:
 08B05 Equational logic, Mal’tsev conditions 20N05 Loops, quasigroups
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