Published in Volume XXVII, Issue 1, 2017, pages 1-18, doi: 10.7561/SACS.2017.1.1

Authors: J.A. Bergstra, I. Bethke


Meadows—commutative rings equipped with a total inversion operation—can be axiomatized by purely equational means. We study subvarieties of the variety of meadows obtained by extending the equational theory and expanding the signature.

Full Text (PDF)


[1] J.A. Bergstra and I. Bethke. A negative result on the algebraic specifications of the meadow of rational numbers. arXiv:1507.00548 [math.RA], 2016.

[2] J.A. Bergstra, I. Bethke, and A. Ponse. Cancellation Meadows: A Generic Basis Theorem and Some Applications. The Computer Journal, 56(1):3–14, 2013 doi:10.1093/comjnl/bxs028.

[3] J.A. Bergstra, I. Bethke, and A. Ponse. Equations for formally real meadows. Journal of Applied Logic, 13(2):1–23, 2015. doi:10.1016/j.jal.2015.01.004.

[4] J.A. Bergstra and C.A. Middelburg. Inversive meadows and divisive meadows. Journal of Applied Logic, 9(3):203–220, 2011. doi:10.1016/j.jal.2011.03.001.

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

[6] I. Bethke and P. Rodenburg. The initial meadows. Journal of Symbolic Logic, 75(3): 888–895, 2010. doi:10.2178/jsl/1278682205.

[7] I. Bethke, P. Rodenburg, and A. 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.

[8] D. Bjørner and M.C. Henson (eds). Logics of Specification Languages. Monographs in Theoretical Computer Science, an EATCS Series. Springer, 2007. 10.1007/978-3-540-74107-7.

[9] S. Burris and H.P. Sankappanavar. A Course in Universal Algebra. Springer Verlag, 1981.

[10] J.A. Goguen, J.W. Thatcher, E.G. Wagner, and J.B. Wright. Initial algebra semantics and continuous algebras. Journal of the ACM, 24(1):68–95, 1977. 10.1145/321992.321997.

[11] G. Grätzer. Universal Algebra (2nd ed.), Springer-Verlag, 1979. 10.1007/978-0-387-77487-9.

[12] T.E. Hall. Identities for existence varieties of regular semigroups. Bulletin of the Australian Mathematical Society, 40(1):59–77, 1989. 10.1017/S000497270000349X.

[13] Y. Komori. Free algebras over all fields and pseudo-fields. Reports of Faculty of Science, Shizuoka University, 10:9–15, 1975. 10.14945/00008858.

[14] R.N. McKenzie, G.F. Mc Nulty, and W.F. Taylor. Algebras, lattices, varieties, Wadsworth & Brooks, Monterey, California, 1987.

[15] A. Robinson. Complete theories. Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam-New York- Oxford, 1977.

[16] J. von Neumann. On regular rings. Proceedings of the National Academy of Sciences of the United States of America, 22(12):707–712, 1936.

[17] W. Wechler. Universal Algebra for Computer Scientists. Monographs in Theoretical Computer Science, an EATCS Series. Springer, 1992. 10.1007/978-3-642-76771-5.

[18] M. Wirsing. Algebraic Specification. In J. van Leeuwen (ed.), Handbook of Theretical Computer Science, Vol. B (Formal Models and Semantics), pp. 675–788. Elsevier, 1990.


  title={Subvarieties of the Variety of Meadows},
  author={J.A. Bergstra and I. Bethke},
  journal={Scientific Annals of Computer Science},
  organization={``A.I. Cuza'' University, Iasi, Romania},
  publisher={``A.I. Cuza'' University Press}