Abstract

A foundational algebra ( , f, ) consists of a hemimorphism f on a Boolean algebra with a greatest solution to the condition f(x). The quasi-variety of foundational algebras has a decidable equational theory, and generates the same variety as the complex algebras of structures (X, R), where f is given by R-images and is the non-wellfounded part of binary relation R.The corresponding results hold for algebras satisfying =0, with respect to complex algebras of wellfounded binary relations. These algebras, however, generate the variety of all ( ,f) with f a hemimorphism on ).