Factorable Congruences and Strict Refinement

We show that universal algebras with factorable congruences such as rings with 1 and semirings with 0 and 1 enjoy some of the properties of universal algebras whose congruence lattices are distributive, such as the strict refinement property and a variant of Jónsson’s lemma. A universal algebra A is said to have factorable congruences if whenever A ∼= B × C and θ is a congruence on A, then θ = φ × ψ, where φ is a congruence on B and ψ is a congruence on C (((b, c) , (b′, c′)) ∈ φ × ψ iff (b, b′) ∈ φ and (c, c) ∈ ψ). In other words, if A = B × C and f is a homomorphism of A onto a universal algebra D, there are homomorphisms g of B onto a universal algebra E and h of C onto a universal algebra F such that D ∼= E × F and f = i ◦ (g, h), where i is an isomorphism of E × F onto D; i.e., the operations of forming homomorphic images and finite Cartesian products permute. Varieties of universal algebras in which every algebra has factorable congruences are characterized via Mal’cev conditions by Fraser and Horn [9]. These varieties include varieties whose congruence lattices are distributive such as lattices and not necessarily associative Boolean rings [13], and varieties with uniform congruence schemes [10], [5]. Any algebra with binary terms +, · and constants 0, 1 satisfying the identities x · 1 = x+ 0 = 0 + x = x, x · 0 = 0 has factorable congruences [9]. This includes rings with 1, near-rings with 1, and semirings with 0 and 1. We prove that universal algebras with factorable congruences enjoy the strict refinement property and every directly indecomposable homomorphic image of a Cartesian product of a family K of algebras with factorable congruences is a homomorphic image of an ultraproduct of K. A variety of universal algebras V is said to have the Apple property [2], if the congruence lattice of every finite directly indecomposable member of V has a unique coatom. In [2], Berman and Blok prove a number of theorems concerning finite universal algebras with factorable congruences and satisfying the Apple property. We will show here that some of these theorems hold Received January 18, 1996. 1980 Mathematics Subject Classification (1991 Revision). Primary 08B10; Secondary 08C10.