Skew-field of trace-preserving endomorphisms, of translation group in affine plane

• Zaka, Orgest ^[1] ; Mohammed, Mohanad A. ^[2]
  1. [1] Agricultural University of Tirana

     Agricultural University of Tirana

     Albania

     
  2. [2] Open Educational College.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 39, Nº. Extra 4, 2020 (Ejemplar dedicado a: Special Issue: Mathematical Computation in Combinatorics and Graph Theory; i), págs. 835-850
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-2020-04-0052
• Enlaces
  • Texto completo
• Resumen

  • We will show how to constructed an Skew-Field with trace-preserving endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of endomorphisms algebra and trace-preserving endomorphisms algebra in an affine plane, and we have constructed an associative unitary ring for which trace-preserving endomorphisms. In this paper we formulate and prove an important Lemma, which enables us to construct a particular trace-preserving endomorphism, with the help of which we can construct the inverse trace-preserving endomorphisms of every trace-preserving endomorphism. At the end of this paper we have proven that the set of tracepreserving endomorphisms together with the actions of ’addition’ and ’composition’ (which is in the role of ’multiplication’) forms a skewfield.

• Referencias bibliográficas
  • E. Artin, Geometric algebra. New York, NY: Wiley Interscience, 1988, doi: 10.1002/9781118164518
  • M. Berger, Geometry, vol. 1. Berlin: Springer, 1987, doi: 10.1007/978-3-540-93815-6
  • M. Berger, Geometry, vol. 2. Berlin: Springer, 1987, doi: 10.1007/978-3-540-93816-3
  • H. S. M. Coxeter, Introduction to geometry, 2nd ed. New York, NY: Wiley, 1989.
  • K. Filipi, O. Zaka, and A. Jusufi, “The construction of a corp in the set of points in a line of desargues affine plane”, Matemati?ki bilten,...
  • R. C. Hartshorne, Foundations of projective geometry. New York, NY: Benjamin/Cummings, 1967.
  • D. Hilbert, The foundations of geometry, La Salle, IL: The Open Court Publishing, 1959.
  • D. R. Hughes and F.C. Piper, Projective planes. Berlin: Spriger, 1973
  • A. Kryftis, “Aconstructive approach to affine and projective planes”, Jan. 2016. arXiv: 1601.04998v1
  • S. Lang, Algebra, 3rd ed. New York, NY: Springer, doi: 10.1007/978-1-4613-0041-0
  • J. J. Rotman, Advanced modern algebra, 2nd ed. Providence, RI: American mathematical society., 2010.
  • R. Wisbauer, Foundations of module and ring theory. Philadelphia, PA: Gordon and Breach Science Publishers, 1991.
  • O. Zaka, "Contribution to reports of some algebraic structures with affine plane geometry and applications", Ph. D thesis, Polytechnic...
  • O. Zaka, “Three vertex and parallelograms in the affine plane: similarity and addition abelian groups of similarly n-vertexes in the Desargues...
  • O. Zaka, “Dilations of line in itself as the automorphism of the skew-field constructed over in the same line in Desargues affine plane”,...
  • O. Zaka and K. Filipi, “The transform of a line of Desargues affine plane in an additive group of its points”, International journal. of current...
  • O. Zaka and J.F. Peters, “Isomorphic-dilations of the skew-fields constructed over parallel lines in the desargues affine plane”, Mar. 2019,...
  • O. Zaka and J.F. Peters, “Ordered line and skew-fields in the Desargues affine plane”, May 2019, arXiv:1905.03859
  • O. Zaka, “One construction of an affine plane over a corps”, Journal of advances in mathematics, vol. 12, no. 5, pp. 6200-6206, 2016, doi:...
  • O. Zaka, “A description of collineations-groups of an affine plane”, Libertas mathematica, vol. 37, no. 2, pp. 81-96, 2017. [On line]. Available:...
  • O. Zaka, “The endomorphisms algebra of translations group and associative unitary ring of trace-preserving endomorphisms in affine plane”,...