Abstract
We prove that iteration theories can be introduced as algebras for the monad 
on the category of signatures assigning to every signature 
the rational-
-tree signature. This supports the result that iteration theories axiomatize precisely the equational properties of least fixed points in domain theory: 
is the monad of free rational theories and every free rational theory has a continuous completion.
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References
Adámek, J., Milius, S., Velebil, J.: Elgot Algebras. Logical Methods in Computer Science 2(5:4), 1–31 (2006)
Bénabou, J.: Structures algébriques dans les catégories. Cah. Topol. Géom. Différ. Catég. 10, 1–126 (1968)
Bloom, S.L., Ésik, Z.: Iteration Theories. Springer, Heidelberg (1993)
Bloom, S.L., Ésik, Z.: Fixed-point operations on ccc’s, Part I. Theoret. Comput. Sci. 155, 1–38 (1996)
Bloom, S.L., Ésik, Z.: The equational logic of fixed points. Theoret. Comput. Sci. 179, 1–60 (1997)
Ésik, Z.: Axiomatizing iteration categories. Acta Cybernetica 14, 65–82 (1999)
Ginali, S.: Regular trees and the free iterative theory. J. Comput. Syst. Sci. 18, 228–242 (1979)
Kelly, G.M., Power, J.: Adjunctions whose units are coequalizers and presentations of finitary enriched monads. J. Pure Appl. Algebra 89, 163–179 (1993)
MacLane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Heidelberg (1998)
Simpson, A., Plotkin, G.: Complete axioms for categorical fixed-point operators. In: IEEE Symposium Logic in Computer Science, pp. 30–41 (1998)
Wright, J.B., Thatcher, J.W., Wagner, E.G., Goguen, J.A.: Rational algebraic theories and fixed-point solutions. In: Proc. 17th IEEE Symposium on Foundations of Computing, Houston, Texas, pp. 147–158. IEEE Computer Society Press, Los Alamitos (1976)
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Adámek, J., Milius, S., Velebil, J. (2007). What Are Iteration Theories?. In: Kučera, L., Kučera, A. (eds) Mathematical Foundations of Computer Science 2007. MFCS 2007. Lecture Notes in Computer Science, vol 4708. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74456-6_23
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DOI: https://doi.org/10.1007/978-3-540-74456-6_23
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