For a variety with weak existentially definable factor congruences, we characterize when the properties “[Formula: see text] is a central element” and “[Formula: see text] and [Formula: see text] are complementary central elements” are definable by [Formula: see text]-formulas and by [Formula: see text]-formulas.
We give a short proof of the fundamental theorem of central element theory (see: Sanchez Terraf and Vaggione, Varieties with definable factor congruences, T.A.M.S. 361). The original proof is constructive and very involved and relies strongly on the fact that the class be a variety. Here we give a more direct nonconstructive proof which applies for the more general case of a first-order class which is both closed under the formation of direct products and direct factors.
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