Dual Number Meadows
DOI:
https://doi.org/10.36285/tm.v0i0.11Abstract
The class of dual number meadows is introduced. By definition this class is a quasivariety. Dual number meadows contain a non-zero element the square of which is zero. These structures are non-involutive and coregular. Some properties of the equational theory of dual number meadows are discussed and an initial algebra specification is given for the minimal dual number meadow of characteristic zero which contains the dual rational numbers. Several open problems are stated.
References
J.A. Anderson, N. Völker, and A. A. Adams. Perspex Machine VIII: axioms of transreal arithmetic. Vision Geometry XV, eds. J. Latecki, D. M. Mountand A. Y. Wu, 649902, (2007).
J.A. Bergstra, I. Bethke. Note on paraconsistency and reasoning about fractions. J. of Applied Non-Classical Logics, (2015).
J.A. Bergstra, I. Bethke. Subvarieties of the variety of meadows. Scientific Annals of Computer Science, vol 27 (1) 1–18, (2017).
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).
J. A. Bergstra, I. Bethke, and A. Ponse. Equations for formally real meadows. Journal of Applied Logic, 13 (2): 1–23, (2015).
J.A. Bergstra, Y. Hirshfeld, and J.V. Tucker. Meadows and the equational specification of division. Theoretical Computer Science, 410 (12), 1261–1271 (2009).
J.A. Bergstra and C.A. Middelburg. Inversive meadows and divisive meadows. Journal of Applied Logic, 9 (3): 203–220 (2011).
J.A. Bergstra and C.A. Middelburg. Division by zero in non-involutive meadows. Journal of Applied Logic, 13 (1): 1–12 (2015).
J.A. Bergstra and A. Ponse. Division by zero in common meadows. In R. de Nicola and R. Hennicker (editors), Software, Services, and Systems (Wirsing Festschrift), LNCS 8950, pages 46-61, Springer, (2015).
J.A. Bergstra and J.V. Tucker. The rational numbers as an abstract datatype. Journal of the ACM, 54 (2), Article 7 (2007).
I. Bethke and P.H. Rodenburg. The initial meadows. Journal of Symbolic Logic, 75 (3), 888-895, (2010).
V. Brodsky and M. Shoham. Dual numbers representation of rigid body dynamics. Mechanism and Machine Theory, 34, 693-718, (1999).
H. D. Ehrich, M. Wolf, and J. Loeckx. Specification of Abstract Data Types. Vieweg + Teubner, ISBN-10:3519021153, (1997)
Y. Komori. Free algebras over all fields and pseudo-fields. Report 10, pp. 9-15, Faculty of Science, Shizuoka University (1975).
H. Michiwaki, S. Saitoh, and N. Yamada. Reality of the division by zero z/0 = 0. International Journal of Applied Physics and mathematics, DOI:10.17706/ijapm.2015.5.3.185-191 (2016).
T.S. dos Reis and J.A.D.W. Anderson. Construction of the transcomplex numbers from the complex numbers. Proc. WCECS 2014, pp 97-102, (2014).
H. Ono. Equational theories and universal theories of fields. Journal of the Mathematical Society of Japan, 35(2), 289-306 (1983).
G.R. Veldkamp. On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics. Mechanism and Machine Theory, 11, 141-156 (1976).
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