Varieties with definable factor congruences

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 $\mathcal{V}$ has DFC if and only if $\mathcal{V}$ has $\vec{0}$ & $\vec{1}$ and Boolean Factor Congruences. We also obtain an explicit first-order definition $\Phi$ 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.