An algebraic study of well-foundedness

✍ Scribed by Robert Goldblatt

• Book ID: 104744903
• Publisher: Springer Netherlands
• Year: 1985
• Tongue: English
• Weight: 948 KB
• Volume: 44
• Category: Article
• ISSN: 0039-3215
• DOI: 10.1007/bf00370431

✦ Synopsis

A foundational algebra (~B, f, A) consists of a heinimorphism f on a Boolean algebra ~B with a greatest solution ~ tO the condition x < 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,/~), where f is given by ~-iinages and A is the non-wellfounded part of binary relation/~.

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 ~.

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