An algebraic study of well-foundedness

Abstract

A foundational algebra ($\mathfrak{B}$, f, λ) consists of a hemimorphism f on a Boolean algebra$\mathfrak{B}$ with a greatest solution λ to the condition α⩽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 λ is the non-wellfounded part of binary relation R.

The corresponding results hold for algebras satisfying λ=0, with respect to complex algebras of wellfounded binary relations. These algebras, however, generate the variety of all ($\mathfrak{B}$,f) with f a hemimorphism on $\mathfrak{B}$).

Admitting a second hemimorphism corresponding to the transitive closure of R allows foundational algebras to be equationally defined, in a way that gives a refined analysis of the notion of diagonalisable algebra.