Distributive Extensions and Quasi-Framal Algebras

Abstract

In (2; 3; 4), A. L. Foster denned Boolean extensions of framal algebras and bounded Boolean extensions of framal-in-the-small algebras. Foster proved that the class of Boolean (of bounded Boolean) extensions of a framal (a framal-in-the-small) algebra A is coextensive up to isomorphism with a certain class of subdirect powers of A, namely, the class of normal (of bounded normal) subdirect powers of A. His proofs apply, however, to considerably more general situations. Indeed, as remarked in (2), the construction of Boolean extensions may be carried out for an arbitrary universal algebra with finitary operations; this is done, in fact, in (4).