Published in Volume XXXII, Issue 1, 2022, pages 87-107, doi: 10.7561/SACS.2022.1.87

Authors: J.A. Bergstra, J.V. Tucker


The formal theory of division in arithmetical algebras reconstructs fractions as syntactic objects called fracterms. Basic to calculation, is the simplification of fracterms to fracterms with one division operator, a process called fracterm attening. We consider the equational axioms of a calculus for calculating with fracterms to determine what is necessary and sufficient for the fracterm calculus to allow fracterm flattening. For computation, arithmetical algebras require operators to be total for which there are several semantical methods. It is shown under what constraints up to isomorphism, the unique total and minimal enlargement of a field Q(\div) of rational numbers equipped with a partial division operator \div has fracterm attening.

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[1] James Anderson and Jan Bergstra. Review of Suppes 1957 proposals for division by zero. Transmathematica, 2021. doi:10.36285/tm.53.

[2] James A. D. W. Anderson, Norbert Volker, and Andrew A. Adams. Perspex Machine VIII: axioms of transreal arithmetic. In Longin Jan Latecki, David M. Mount, and Angela Y.Wu, editors, Vision Geometry XV, volume 6499, pages 7 -18. International Society for Optics and Photonics, SPIE, 2007. doi:10.1117/12.698153.

[3] Jan A. Bergstra. Division by zero: a survey of options. Transmathematica, 2019. doi:10.36285/tm.v0i0.17.

[4] Jan A. Bergstra. Arithmetical datatypes, fracterms, and the fraction definition problem. Transmathematica, 2020. doi:10.36285/tm.33.

[5] Jan A. Bergstra. Fractions in transrational arithmetic. Transmathematica, 2020. doi:10.36285/tm.19.

[6] Jan A. Bergstra and Cornelis A. Middelburg. Transformation of fractions into simple fractions in divisive meadows. Journal of Applied Logic, 16:92-110, 2016. doi:10.1016/j.jal.2016.03.001.

[7] Jan A. Bergstra and Alban Ponse. Division by zero in common meadows. In Rocco De Nicola and Rolf Hennicker, editors, Software, Services, and Systems – Essays Dedicated to Martin Wirsing on the Occasion of His Retirement from the Chair of Programming and Software Engineering, volume 8950 of Lecture Notes in Computer Science, pages 46-61. Springer, 2015. doi:10.1007/978-3-319-15545-6\_6.

[8] Jan A. Bergstra and Alban Ponse. Fracpairs and fractions over a reduced commutative ring. Indagationes Mathematicae, 27(3):727-748, 2016. doi:10.1016/j.indag.2016.01.007.

[9] Jan A. Bergstra and Alban Ponse. Division by zero in common meadows, version 4, modified (stronger) version of [7], 2021. arXiv: 1406.6878v4.

[10] Jan A. Bergstra and John V. Tucker. The rational numbers as an abstract data type. Journal of the ACM, 54(2):7, 2007. doi:10.1145/1219092.1219095.

[11] Jan A. Bergstra and John V. Tucker. The transrational numbers as an abstract data type. Transmathematica, 2020. doi:10.36285/tm.47.

[12] Jan A. Bergstra and John V. Tucker. The wheel of rational numbers as an abstract data type. In Markus Roggenbach, editor, 25th International Workshop on Recent Trends in Algebraic Development Techniques, WADT 2020, Virtual Event, Revised Selected Papers, volume 12669 of Lecture Notes in Computer Science, pages 13-30. Springer, 2020. doi:10.1007/978-3-030-73785-6\_2.

[13] Jan A Bergstra and John V Tucker. On the axioms of common meadows: Fracterm calculus, flattening and incompleteness. The Computer Journal, 2022. doi:10.1093/comjnl/bxac026.

[14] Jan A. Bergstra and John V. Tucker. Totalising partial algebras: Teams and splinters. Transmathematica, 2022. doi:10.36285/tm.57.

[15] Inge Bethke, Piet Rodenburg, and Arjen Sevenster. The structure of finite meadows. Journal of Logical and Algebraic Methods in Programming, 84(2):276-282, 2015. doi:10.1016/j.jlamp.2014.08.004.

[16] Jesper Carlstrom. Wheels – on division by zero. Mathematical Structures in Computer Science, 14(1):143-184, 2004. doi:10.1017/S0960129503004110.

[17] Jacques Loeckx, Hans-Dieter Ehrich, and Markus Wolf. Specification of abstract data types. Wiley, 1996.

[18] Judah P. Makonye and Duduzile Winnie Khanyile. Probing grade 10 students about their mathematical errors on simplifying algebraic fractions. Research in Education, 94(1):55-70, 2015. doi:10.7227/RIE.0022.

[19] Karl Meinke and John V. Tucker. Universal algebra. In Samson Abramsky and Tom S. E. Maibaum, editors, Handbook of Logic in Computer Science (Vol. 1): Background: Mathematical Structures, page 189-368. Oxford University Press, Inc., USA, 1993.

[20] Tiago Reis, Walter Gomide, and James Anderson. Construction of the transreal numbers and algebraic transfields. IAENG International Journal of Applied Mathematics, 46(1):11-23, 2016.

[21] Anton Setzer. Wheels (draft). 1997. URL:

[22] David R. Stoutemyer. Ten commandments for good default expression simplification. Journal of Symbolic Computation, 46(7):859-887, 2011. doi:10.1016/j.jsc.2010.08.017.


  title={Which Arithmetical Data Types Admit Fracterm Flattening?},
  author={J.A. Bergstra, J.V. Tucker},
  journal={Scientific Annals of Computer Science},
  organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania},
  publisher={Alexandru Ioan Cuza University Press, Ia\c si},