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Notions of iteration range from the arguably most general Elgot iteration toa very specific Kleene iteration. The fundamental nature of Elgot iteration hasbeen extensively explored by Bloom and Esik in the form of iteration theories,while Kleene iteration became extremely popular as an integral part of(untyped) formalisms, such as automata theory, regular expressions and Kleenealgebra. Here, we establish a formal connection between Elgot iteration andKleene iteration in the form of Elgot monads and Kleene monads, respectively.We also introduce a novel class of while-monads, which like Kleene monads admita relatively simple description in algebraic terms. Like Elgot monads,while-monads cover a large variety of models that meaningfully supportwhile-loops, but may fail the Kleene algebra laws, or even fail to support aKleen iteration operator altogether.

Shades of Iteration: from Elgot to Kleene