Abstract
A category of geometric spaces (non-commutative geometries) is presented here in an overview style with the objective to propose it as a field for interdisciplinary work in geometric reasoning and symbolic computation under the aspect of interactions of algebraic and geometric approaches. The problem of automated verification of geometric conditions is discussed. Possible directions of further work are included as a prospect.
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References
J. André and H. Ney, On Anshel-Clay-nearrings,Proc. Int. Conf. Nearrings and Nearfields, Oberwolfach, 1991.
J. André, Lineare Algebra über Fastkörpern, Math. Z. 136(1974)295–313.
J. André, On finite noncommutative affine spaces, in:Combinatorics, 2nd Ed., eds. M. Hall, Jr. and J.H. van Lint, Proc. of an Advanced Inst. on Combinat., Breukelen (Amsterdam Math. Centrum, 1974) pp. 65–113.
J. André, Eine geometrische Kennzeichnung imprimitiver Frobenius Gruppen, Abhandlungen Math. Seminar Univ. Hamburg 51(1981)120–135.
J. André, Endliche nichtkommutative Geometrie (Lecture Notes), Ann. Univ. Saraviensis (Ser. Math.) 2(1988)1–136.
J. André, Configurational conditions and digraphs, J. Geom. 43(1992)22–29.
B. Buchberger, Gröbner bases: An algorithmic method in polynomial ideal theory,Multidimensional Systems Theory, ed. N.K. Bose (Reidel, Dordrecht/Boston/Lancaster, 1985) pp. 184–232.
G.E. Collins, Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition, Springer Lecture Notes in Computer Science 33 (1975).
W. Dicks and M.J. Dunwoody,Groups Acting on Graphs (Cambridge University Press, Cambridge, 1988).
P. Fuchs, G. Hofer and G. Pilz, Codes from planar near rings, IEEE Trans. Inform. Theory 36(1990)647–651.
U. Höfer, Constructive methods in finitek-tactical geometric spaces, Diploma Thesis, Saarbrücken (1985), in German.
H. Hong, Improvements in c.a.d.-based quantifier elimination (Ph.D. Thesis), Technical Report, The Ohio State University, Columbus, Ohio (1990).
A. Kerber,Algebraic Combinatorics Via Finite Group Actions (BI Wissenschaftsverlag, Mannheim/Wien/Zürich, 1991).
S. MacLane and I. Moerdijk,Sheaves in Geometry and Logic (Springer, Universitext, 1992).
J. Pfalzgraf, On a model for noncommutative geometric spaces, J. Geom. 25(1985)147–163.
J. Pfalzgraf, On geometries associated with group operations, Geom. Dedicata 21(1986)193–203.
J. Pfalzgraf, A note on simplices as geometric configurations, Archiv der Math. 49(1987)134–140.
J. Pfalzgraf, Representation of geometric spaces as fibered structures, Results in Math. 12(1987)172–190, in German.
J. Pfalzgraf, On graph products of groups and group spaces, in preparation.
G. Pilz,Near-rings, 2nd Ed. (North-Holland, Amsterdam, 1983).
D.E. Rydeheard and R.M. Burstall,Computational Category Theory, Ser. in Comp. Sci. (Prentice-Hall, 1988).
J.P. Serre,Trees (Springer, Berlin/Heidelberg/New York, 1980).
A. Tarski,A Decision Method for Elementary Algebra and Geometry (The RAND Corporation, Santa Monica; 2nd Ed., University of California Press, Berkeley and Los Angeles, 1951).
D.M. Wang, Reasoning about geometric problems using algebraic methods,Proc. MEDLAR 24-Month Review Workshop, Grenoble, 1991, RISC-Linz Ser. No. 91.51.0, 1991.
D.M. Wang, Solution to a benchmark example, private communication (letter) (1993).
W.T. Wu, Basic principles of mechanical theorem proving in elementary geometries, J. Autom. Reasoning 2(1986)221–252.
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Dedicated to Professor Dana Scott on the occasion of his 60th birthday
Sponsored by the Austrian Science Foundation (FWF), ESPRIT BRP 6471 MEDLAR II.
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Pfalzgraf, J. A category of geometric spaces: Some computational aspects. Ann Math Artif Intell 13, 173–193 (1995). https://doi.org/10.1007/BF01531328
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DOI: https://doi.org/10.1007/BF01531328