Congruence Boolean Lifting Property (1502.06907v2)

Abstract: We introduce and study the Congruence Boolean Lifting Property (CBLP) for congruence--distributive universal algebras, as well as a property related to CBLP, which we have called $(\star )$. CBLP extends the so--called Boolean Lifting Properties (BLP) from MV--algebras, BL--algebras and residuated lattices, but differs from the BLP when particularized to bounded distributive lattices. Important classes of universal algebras, such as discriminator varieties, fulfill the CBLP. The main results of the present paper include a characterization theorem for congruence--distributive algebras with CBLP and a structure theorem for semilocal arithmetical algebras with CBLP. When we particularize the CBLP to the class of residuated lattices and to that of bounded distributive lattices and we study its relation to other Boolean Lifting Properties for these algebras, interesting results concerning the image of the reticulation functor between these classes are revealed.