  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 of the kernel of the canonical projections
via the terms associated to the Mal'cev condition for DFC, in such a manner 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.

Logic