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.
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The research reported in this paper was carried out while the author was at the Department of Mathematics and Statistics, University of Auckland, under a teaching exchange with the Victoria University of Wellington.
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Goldblatt, R. An algebraic study of well-foundedness. Stud Logica 44, 423–437 (1985). https://doi.org/10.1007/BF00370431
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DOI: https://doi.org/10.1007/BF00370431