Varieties with Definable Factor Congruences

TL;DR

This paper investigates when algebraic varieties have definable factor congruences, establishing that this property is equivalent to having certain logical and algebraic conditions, and providing explicit first-order definitions related to direct product representations.

Contribution

It proves that Definable Factor Congruences (DFC) is a Mal'cev property and characterizes DFC through equivalent algebraic and logical conditions, including the presence of 0&1 and Boolean factor congruences.

Findings

DFC is a Mal'cev property.

DFC is equivalent to having 0&1 and Boolean factor congruences.

Explicit first-order definitions of kernels are provided.

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 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.