Varieties with definable factor congruences

P. SANCHEZ TERRAF; D. VAGGIONE

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

American Mathematical Society

Año: 2008

0002-9947

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.