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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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