Abstract
First a problem of Ralph McKenzie is answered by proving that in a finitely directly representable variety every directly indecomposable algebra must be finite. Then we show that there is no local proof of the fundamental theorem of Abelian algebras nor of H. P. Gumm's permutability results. This part may also be of interest for those investigating non-modular Abelian algebras. Finally we provide a Gumm-type characterization of the situation when twonot necessarily comparable congruences centralize each other. In doing this, we introduce a four variable version of the difference term in every modular variety. A “two-terms condition” is also investigated.
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