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The initial meadows

Published online by Cambridge University Press:  12 March 2014

Inge Bethke  and
Piet Rodenburg
Inge Bethke
Affiliation:
Informatics Institute, University of Amsterdam, Science Park 107, 1098 Xg Amsterdam, The Netherlands. E-mail: I.Bethke@uva.nl
Piet Rodenburg
Affiliation:
Informatics Institute, University of Amsterdam, Science Park 107, 1098 Xg Amsterdam, The Netherlands. E-mail: P.H.Rodenburg@uva.nl
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Abstract

A meadow is a commutative ring with an inverse operator satisfying 0−1 = 0. We determine the initial algebra of the meadows of characteristic 0 and prove a normal form theorem for it. As an immediate consequence we obtain the decidability of the closed term problem for meadows and the computability of their initial object.

Keywords

Data structuresspecification languagesinitial algebra semanticsword problemdecidabilitycomputable algebrasnormal forms
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 888 - 895
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1]Bergstra, J.A., Hirshfeld, Y., and Tucker, J.V., Fields, meadows and abstract data types, Pillars of computer science, essays dedicated to Boris (Boaz) Trakhtenbrot on the occasion of his 85th birthday (Avron, A.et al, editor), Lecture Notes in Computer Science, vol. 4800, Springer, 2008, pp. 166–178.CrossRefGoogle Scholar
[2]Bergstra, J.A., Hirshfeld, Y., and Tucker, J.V., Meadows and the equational specification of division, Theoretical Computer Science, vol. 410 (2009), pp. 1261–1271.CrossRefGoogle Scholar
[3]Bergstra, J.A. and Tucker, J.V., The rational numbers as an abstract data type, Journal of the ACM, vol. 54 (2007), no. 2.CrossRefGoogle Scholar
[4]Bethke, I. and Rodenburg, P., Some properties of finite meadows, CoRR abs/0712.0917, 2007.Google Scholar
[5]Birkhoff, G., Subdirect unions in universal algebra, Bulletin of the American Mathematical Society, vol. 50 (1944), no. 10, pp. 764–768.CrossRefGoogle Scholar
[6]Birkhoff, G., Lattice theory, vol. 25, American Mathematical Society Colloquium Publications, 1991.Google Scholar
[7]Goguen, J.A., Thatcher, J.W., Wagner, E.G., and Wright, J.B., Initial algebra semantics and continuous algebras. Journal of the ACM, vol. 24 (1977), no. 1, pp. 68–95.CrossRefGoogle Scholar
[8]Goodearl, K.R., Von Neumann regular rings, Pitman, London, San-Francisco, Melbourne, 1979.Google Scholar
[9]Grätzer, G., Universal algebra, 2nd ed., Springer, 1979.CrossRefGoogle Scholar
[10]McKenzie, R.N., Nulty, G.F. Mc, and Taylor, W.F., Algebras, lattices, varieties, Wadsworth & Brooks, Monterey, California, 1987.Google Scholar
[11]Stoltenberg-Hansen, V. and Tucker, J.V., Computable rings and fields, Handbook of computability theory (Griffor, E.R., editor), Studies in Logic and the Foundations of Mathematics, vol. 140, Elsevier, 1999, pp. 363–447.CrossRefGoogle Scholar
[12]Wechler, W., Universal algebra for computer scientists, EATCS Monographs in Computer Science, Springer, 1992.CrossRefGoogle Scholar