Skip to main content

Iterative reflections of monads


Iterative monads were introduced by Calvin Elgot in the 1970's and are those ideal monads in which every guarded system of recursive equations has a unique solution. We prove that every ideal monad has an iterative reflection, that is, an embedding into an iterative monad with the expected universal property. We also introduce the concept of iterativity for algebras for the monad , following in the footsteps of Evelyn Nelson and Jerzy Tiuryn, and prove that is iterative if and only if all free algebras for are iterative algebras.

Hide All
Aczel, P., Adámek, J., Milius, S. and Velebil, J. (2003) Infinite trees and completely iterative theories: a coalgebraic view. Theoretical Computer Science 300 145.
Adámek, J. and Milius, S. (2006) Terminal coalgebras and free iterative theories. Inform. and Comput. 204 11391172.
Adámek, J., Milius, S. and Velebil, J. (2006) Iterative algebras at work. Mathematical Structures in Computer Science 16 (6)10851131.
Adámek, J., Milius, S. and Velebil, J. (2009a) Semantics of higher-order recursion schemes. Proceedings CALCO 2009. Springer-Verlag Lecture Notes in Computer Science 5728 4963.
Adámek, J., Milius, S. and Velebil, J. (2009b) A description of iterative reflections of monads. Extended abstract in Proc. FOSSACS 2009. Springer-Verlag Lecture Notes in Computer Science 5504 152166.
Adámek, J. and Rosický, J. (1994) Locally presentable and accessible categories, Cambridge University Press.
Badouel, E. (1989) Terms and infinite trees as monads over a signature. Springer-Verlag Lecture Notes in Computer Science 351 89103.
Barr, M. (1970) Coequalizers and free triples. Math. Z. 116 307322.
Bloom, S. and Ésik, Z. (1993) Iteration theories: the equational logic of iteration processes, EATCS Monographs on Theoretical Computer Science.
Carboni, A., Lack, S. and Walters, R. F. C. (1993) Introduction to extensive and distributive categories. J. Pure Appl. Algebra 84 145158.
Elgot, C. C. (1975) Monadic computation and iterative algebraic theories. In: Rose, H. E. and Shepherdson, J. C. (eds.) Logic Colloquium '73, North-Holland.
Elgot, C. C., Bloom, S. and Tindell, R. (1978) On the algebraic structure of rooted trees. J. Comput. System Sci. 16 361399.
Fiore, M., Plotkin, G. and Turi, D. (1999) Abstract syntax and variable binding. Proc. Logic in Computer Science 1999, IEEE Press 193–202.
Gabriel, P. and Ulmer, F. (1971) Lokal präsentierbare Kategorien. Springer-Verlag Lecture Notes in Mathematics 221.
Ginali, S. (1979) Regular Trees and the Free Iterative Theory. J. Comput. System Sci. 18 228242.
MacLane, S. (1998) Categories for the working mathematician, 2nd edition, Springer-Verlag.
Nelson, E. (1983) Iterative algebras. Theoretical Computer Science 25 6794.
Tiuryn, J. (1980) Unique fixed points vs. least fixed points. Theoretical Computer Science 12 229254.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed