Elgot Algebras

Abstract

Iterative algebras, i. e., algebras A in which flat recursive equations e have unique solutions $e^{†}$, are generalized to Elgot algebras, where a choice $e\mapsto e^{†}$ of solutions of all such equations e is specified. This specification satisfies two simple and well motivated axioms: functoriality (stating that solutions are “uniform”) and compositionality (stating how to perform simultaneous recursion). These two axioms stem canonically from Elgot's iterative theories: We prove that the category of Elgot algebras is the Eilenberg–Moore category of the free iterative monad.