Abstract
We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0 & 1 and Boolean Factor Congruences. We also obtain an explicit first-order definition Φ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner that it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.
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