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RINGS WITH COMMON DIVISION, COMMON MEADOWS AND THEIR CONDITIONAL EQUATIONAL THEORIES

Part of: Algebraic structures Connections with logic Other classes of algebra Varieties

Published online by Cambridge University Press:  23 December 2024

JAN A. BERGSTRA Open the ORCID record for JAN A. BERGSTRA [Opens in a new window]  and
JOHN V. TUCKER Open the ORCID record for JOHN V. TUCKER [Opens in a new window]
JAN A. BERGSTRA
Affiliation:
INFORMATICS INSTITUTE, UNIVERSITY OF AMSTERDAM SCIENCE PARK 900, 1098 XH, AMSTERDAM THE NETHERLANDS E-mail: j.a.bergstra@uva.nl
JOHN V. TUCKER*
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE SWANSEA UNIVERSITY, BAY CAMPUS FABIAN WAY, SWANSEA, SA1 8EN UK
*
E-mail: j.v.tucker@swansea.ac.uk
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Abstract

We examine the consequences of having a total division operation $\frac {x}{y}$ on commutative rings. We consider two forms of binary division, one derived from a unary inverse, the other defined directly as a general operation; each are made total by setting $1/0$ equal to an error value $\bot $, which is added to the ring. Such totalised divisions we call common divisions. In a field the two forms are equivalent and we have a finite equational axiomatisation E that is complete for the equational theory of fields equipped with common division, which are called common meadows. These equational axioms E turn out to be true of commutative rings with common division but only when defined via inverses. We explore these axioms E and their role in seeking a completeness theorem for the conditional equational theory of common meadows. We prove they are complete for the conditional equational theory of commutative rings with inverse based common division. By adding a new proof rule, we can prove a completeness theorem for the conditional equational theory of common meadows. Although, the equational axioms E fail with common division defined directly, we observe that the direct division does satisfy the equations in E under a new congruence for partial terms called eager equality.

Keywords

commutative ringsdivision operatorsmeadowscommon divisioncommon meadowsequationsconditional equationsvarietiesquasivarieties

MSC classification

Primary: 08B05: Equational logic, Mal′cev (Mal′tsev) conditions 08C15: Quasivarieties 12L12: Model theory
Secondary: 08A55: Partial algebras

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Type
Article
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The Journal of Symbolic Logic , First View , pp. 1 - 31
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

REFERENCES

Anderson, J. A. D. W. and Bergstra, J. A., Review of Suppes 1957 proposals for division by zero . Transmathematica, (2021). https://doi.org/10.36285/tm.53.Google Scholar
Anderson, J. A. D. W., Völker, N., and Adams, A. A.. Perspex Machine VIII, axioms of transreal arithmetic , Proceeding of SPIE 6499, Vision Geometry XV (Latecki, J., Mount, D. M., and Wu, A. Y., editors), 29 January 2007, 649902. https://doi.org/10.1117/12.698153.Google Scholar
Bergstra, J. A., Division by zero, a survey of options . Transmathematica, (2019). https://doi.org/10.36285/tm.v0i0.17.Google Scholar
Bergstra, J. A., Dual number meadows . Transmathematica, (2019). https://doi.org/10.36285/tm.v0i0.11.Google Scholar
Bergstra, J. A., Arithmetical datatypes, fracterms, and the fraction definition problem . Transmathematica, (2020). https://doi.org/10.36285/tm.33.Google Scholar
Bergstra, J. A., Bethke, I., and Ponse, A., Cancellation meadows: a generic basis theorem and some applications . The Computer Journal, vol. 56 (2013), no. 1, pp. 3–14.Google Scholar
Bergstra, J. A. and Middelburg, C. A., Transformation of fractions into simple fractions in divisive meadows . Journal of Applied Logic, vol. 16 (2016), pp. 92–110.Google Scholar
Bergstra, J. A. and Ponse, A.. Division by zero in common meadows , Software, Services, and Systems (Wirsing Festschrift) (de Nicola, R. and Hennicker, R., editors), Lecture Notes in Computer Science, 8950, Springer, Berlin, 2015, pp. 46–61, https://doi.org/10.1007/978-3-319-15545-6_6.Google Scholar
Bergstra, J. A. and Tucker, J. V., The rational numbers as an abstract data type . Journal of the ACM, vol. 54 (2007), no. (2), p. 7 (25 pages).Google Scholar
Bergstra, J. A. and Tucker, J. V., The transrational numbers as an abstract datatype . Transmathematica, (2020). https://doi.org/10.36285/tm.47.Google Scholar
Bergstra, J. A. and Tucker, J. V., The wheel of rational numbers as an abstract data type , Recent Trends in Algebraic Development Techniques. WADT 2020 (Roggenbach, M., editor), Lecture Notes in Computer Science, 12669, Springer, Berlin, 2021, pp. 13–30.Google Scholar
Bergstra, J. A. and Tucker, J. V.. Totalising Partial algebras: teams and splinters . Transmathematica, (2022). https://doi.org/10.36285/tm.57.Google Scholar
Bergstra, J. A. and Tucker, J. V.. On the axioms of common meadows: Fracterm calculus, flattening and incompleteness . The Computer Journal, vol. 66 (2022), no. 7, pp. 1565–1572.Google Scholar
Bergstra, J. A. and Tucker, J. V., Eager equality for rational number arithmetic . ACM Transactions on Computational Logic, vol. 24 (2023), no. 3, Article 22, pp 1–28.Google Scholar
Bergstra, J. A. and Tucker, J. V., Eager term rewriting for the fracterm calculus of common meadows . The Computer Journal, vol. 67 (2024), no. 5, 1866–1871. https://doi.org/10.1093/comjnl/bxad106.Google Scholar
Bergstra, J. A. and Tucker, J. V., Tucker A complete finite axiomatisation of the equational theory of common meadows . ACM Transactions on Computational Logic, vol. 26 (January 2025), no. 1, Article 1, 28. https://doi.org/10.1145/3689211.Google Scholar
Bergstra, J.A. and Tucker, J. V., Symmetric transrationals: The data type and the algorithmic degree of its equational theory , A Journey From Process Algebra via Timed Automata to Model Learning – A Festschrift Dedicated to Frits Vaandrager on the Occasion of His 60th Birthday (Jansen, N. et al., editors), Lecture Notes in Computer Science, 13560, Springer, Berlin, 2022, pp. 63–80.Google Scholar
Birkhoff, G., On the structure of abstract algebras . Mathematical Proceedings of the Cambridge Philosophical Society, vol. 31 (1935), no. 4, pp. 433–454.Google Scholar
Burris, S. and Sankapannavar, H. P., A Course in Universal Algebra, Springer-Verlag, Berlin, 1981.Google Scholar
Carlström, J., Wheels–on division by zero . Mathematical Structures in Computer Science, vol. 14 (2004), no. 1, pp. 143–184.Google Scholar
Dias, J. and Dinis, B., Strolling through common meadows . Communications in Algebra, vol. 52 (2024), no. 12, pp. 5015–5042. Also: arXiv:2311.05460 [math.RA] (2023).Google Scholar
Dias, J. and Dinis, B., Towards an enumeration of finite common meadows . International Journal of Algebra and Computation, vol. 34 (2024), no. 6, pp. 837–855.Google Scholar
dos Reis, T. S., Gomide, W., and Anderson, J. A. D. W., Construction of the transreal numbers and algebraic transfields . IAENG International Journal of Applied Mathematics, vol. 46 (2016), no. 1, pp. 11–23. (http://www.iaeng.org/IJAM/issues_v46/issue_1/IJAM_46_1_03.pdf)Google Scholar
Ehrich, H. D., Wolf, M., and Loeckx, J., Specification of Abstract Data Types, Vieweg Teubner, John Wiley, Chichester, 1997.Google Scholar
Kleene, S. C.. Introduction to Metamathematics, North Holland Publishing Co., Amsterdam, 1952.Google Scholar
Mal’tsev, A. I.. Algebraic Systems, Springer, Berlin, 1973.Google Scholar
Meinke, K. and Tucker, J. V., Universal Algebra , Handbook of Logic in Computer Science (Abramsky, S., Gabbay, D. M., and Maibaum, T. S. E., editors), Oxford University Press, Oxford, 1992. https://doi.org/10.1093/oso/9780198537359.003.0003.Google Scholar
Ono, H., Equational theories and universal theories of fields . Journal of the Mathematical Society of Japan, vol. 35 (1983), no. 2, pp. 289–306.Google Scholar
Setzer, A.. Wheels (draft), 1997. http://www.cs.swan.ac.uk/csetzer/articles/wheel.pdf.Google Scholar
Suppes, P., Introduction to Logic, Van Nostrand Reinhold, Princeton, 1957.Google Scholar
van der Waerden, B. L., Modern Algebra, vol. 1, Frederick Ungar Publishing Company, New York, 1970.Google Scholar