FoSSaCS 2009: Foundations of Software Science and Computational Structures pp 152-166 | Cite as
A Description of Iterative Reflections of Monads (Extended Abstract)
Abstract
For ideal monads in Set (e. g. the finite list monad, the finite bag monad etc.) we have recently proved that every set generates a free iterative algebra. This gives rise to a new monad. We prove now that this monad is iterative in the sense of Calvin Elgot, in fact, this is the iterative reflection of the given ideal monad. This shows how to freely add unique solutions of recursive equations to a given algebraic theory. Examples: the monad of free commutative binary algebras has the monad of binary rational unordered trees as iterative reflection, and the finite list monad has the iterative reflection given by adding an absorbing element.
References
- 1.Adámek, J., Börger, R., Milius, S., Velebil, J.: Iterative algebras: How iterative are they? Theory Appl. Categ. 19, 61–92 (2008)MathSciNetMATHGoogle Scholar
- 2.Adámek, J., Milius, S.: Terminal coalgebras and free iterative Theories. Inform. and Comput. 204, 1139–1172 (2006)MathSciNetCrossRefMATHGoogle Scholar
- 3.Adámek, J., Milius, S., Velebil, J.: Iterative algebras at work. Math. Structures Comput. Sci. 16.6, 1085–1131 (2006)MathSciNetCrossRefMATHGoogle Scholar
- 4.Adámek, J., Milius, S., Velebil, J.: Elgot algebras. Logical Methods Comput. Sci. 2(5:4), 31 (2006)MathSciNetMATHGoogle Scholar
- 5.Adámek, J., Milius, S., Velebil, J.: Iterative reflections of monads. Math. Structures Comput. Sci. (accepted for publication)Google Scholar
- 6.Adámek, J., Milius, S., Velebil, J.: A description of iterative reflections of monads, http://www.stefan-milius.eu
- 7.Adámek, J., Rosický, J.: Locally presentable and accessible categories. Cambridge University Press, Cambridge (1994)CrossRefMATHGoogle Scholar
- 8.Barr, M.: Coequalizers and free triples. Math. Z. 116, 307–322 (1970)MathSciNetCrossRefMATHGoogle Scholar
- 9.Elgot, C.C.: Monadic computation and iterative algebraic theories. In: Rose, H.E., Shepherdson, J.C. (eds.) Logic Colloquium 1973. North-Holland Publishers, Amsterdam (1975)Google Scholar
- 10.Elgot, C.C., Bloom, S., Tindell, R.: On the algebraic structure of rooted trees. J. Comput. System Sci. 16, 361–399 (1978)MathSciNetCrossRefMATHGoogle Scholar
- 11.Gabriel, P., Ulmer, F.: Lokal präsentierbare Kategorien. Lecture N. Math., vol. 221. Springer, Berlin (1971)MATHGoogle Scholar
- 12.Ginali, S.: Regular trees and the free iterative theory. J. Comput. System Sci. 18, 228–242 (1979)MathSciNetCrossRefMATHGoogle Scholar
- 13.Mac Lane, S.: Categories for the working mathematician, 2nd edn. Springer, Heidelberg (1998)MATHGoogle Scholar
- 14.Nelson, E.: Iterative algebras. Theoret. Comput. Sci. 25, 67–94 (1983)MathSciNetMATHGoogle Scholar
- 15.Tiuryn, J.: Unique fixed points vs. least fixed points. Theoret. Comput. Sci. 12, 229–254 (1980)MathSciNetCrossRefMATHGoogle Scholar