Gumm's Theorem and the structure of minimal algebras

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In this note we will show that Gumm’s theorem on Abelian algebras can be strengthened in such a way that Palfy’s theorem on minimal algebras becomes a near consequence. This is surprising, at least after having a second look at the matter. For if one compares the proofs of the original theorems there is indeed a similarity between the two; however, it is not possible to see directly that a minimal algebra with more than 3 elements is Abelian. This results only after the completion of the classification of minimal algebras. So, Gumm’s theorem is of no use here. Instead, there is a weaker property that can easily be derived for minimal algebras, which we call 1-Abelian; and we can show that Gumm’s theorem can be proved with 1-Abelian replacing Abelian. It should be said here that few of the proofs are new; rather, the novelty is the arrangement of the facts that can be proved with them.