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Remarks on Two-By-Two Matric Semigroups

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For Dirk Struik

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 15))

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Abstract

This is an identification of many of the important subsets of the set of 2x2 matrices with complex number elements, having under matrix multiplication a semigroup structure. The concepts defined are important and the structures noted are excellent examples for use in mathematics education. Some applications are noted in the references. Since we are concerned only with 2x2 matrices the results are easily verified and the proofs are omitted.

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Bibliography

  • Brown, Dennison E., ‘On Clans of Non-Negative Matrices’, Proc. Amer. Math. Soc. 15 (1964) 671–674.

    Article  Google Scholar 

  • Brown, Dennison E., ‘Matrix Representations of Compact Simple Semigroups’, Duke Math. J. 33 (1966) 69–74.

    Article  Google Scholar 

  • Cheng, Tseng-Tsung, ‘Generalization of De Moivre’s and Fourir’s Theorems to Matrices’, Collect Papers Sci. Engin Nat. Univ. Amoy. 1 (1943) 65–58. MR 8 p. 432.

    Google Scholar 

  • Cohen, Haskell and Collins, H. S., ‘Affine Semigroups’, Trans. Amer. Math. Soc. 93 (1959) 97–113.

    Article  Google Scholar 

  • Fusch-Rabinowitsch, D.J., ‘On a Certain Representation of a Free Group’, Leningrad State Univ. Annals Math Ser. 10 (1940) 154–157; MR 2 p. 215.

    Google Scholar 

  • Gluskin, L.M., ‘Matricial Semigroups?’, Izv. Akad Nauk SSSRS er. Mat. 22 (1958) 439–448; MR 20# 2386.

    Google Scholar 

  • Gluskin, L. M., ‘Matricial Semigroups?’, Izv. Akad Nauk SSSRS er. Mat. 22 (1958) 439–448; MR 20 # 2386.

    Google Scholar 

  • Hille, Einar, ‘What is a Semi-Group?’, Studies in Real and Complex Analysis, 1.1. Hirschman, Jr. (ed.) Studies in Mathematics, Vol. 3, Mathematics Association of America, Prentice Hall, Englewood Cliffs, N. J. 1965, pp. 55–66.

    Google Scholar 

  • Houle, J. E., ‘The Quaternion-Reductibility of Semigroups of Complex Matrices’, Nieuw Arch. Wisk. 8 (1960) 17–21.

    Google Scholar 

  • Houle, J. E., ‘Finite Groups of Quaternion Matrices’, Duke Math. J. 28 (1961) 383–386; MR 24 # A155.

    Google Scholar 

  • Jacobson, Bernard and Wisner, Robert J., ‘Matrix Number Theory: An Example of Non-Unique Factorization’, Amer. Math. Monthly 72 (1965) 399–402.

    Article  Google Scholar 

  • Jakobson, Bernard and Wisner, Robert J., ‘Matrix Number Theory. I: Factorization of 2x2 Unimodular Matrices’, Publ. Math. Debrecen 13 (1966) 67–72.

    Google Scholar 

  • Jacobson, Bernard and Wisner, Robert J., ‘Matrix Number Theory’. II: ‘Factorization of 2 X 2 Singular Matrices’, Amer. Math. Soc. Notices 10 (1963) 264.

    Google Scholar 

  • Kneser, Martin and Puppe, Dieter, ‘Quadratische Formen und Verschlingungsinvarianten von Knoten’ Math Z 58 (1953) 376–384; MR 15 p. 100.

    Google Scholar 

  • Morita, Kitti, ‘Uber normale antilineare Transformationen’, Proc. Imp. Acad. Tokyo 20 (1944) 715–720; MR 7 p. 358.

    Google Scholar 

  • Mulcrone, T. F., S. J., ‘Semigroup Examples in Introductory Modern Algebra’, Amer. Math. Monthly 69 (1962) 296–301.

    Article  Google Scholar 

  • Robinson, D. W., ‘A Note on a Simple Matrix Isomorphism’, Math. Mag. 32 (1958/ 59) 213–215; MR 21 # 3440.

    Google Scholar 

  • Serre, Jean-Pierre, ‘Le problème des groupes de congruence pour SL2’, Ann. Math. (2) 92 (1970) 489–527.

    Article  Google Scholar 

  • Sinkhorn, Richard, ‘Linear Transformations under which the Doubly Stochastic Matrices are Invariant’, Proc. Amer. Math. Soc. 27 (1971) 213–221.

    Article  Google Scholar 

  • Taussky, Olga, ‘Sums of Squares’, Amer. Math. Monthly 77 (1970) 805–830.

    Article  Google Scholar 

  • Thompson, R. C., ‘Unimodular Group Matrices with Rational Integers as Elements’, Pacific J. Math. 14 (1964) 719–726.

    Google Scholar 

  • Vandiver, H. S., ‘Note on an Associative Distributive Algebra in which the Commutative Law of Addition Does not Hold’, Bull. Amer. Math. Soc. 42 (1936) 857–859.

    Article  Google Scholar 

  • Wallace, A. D., ‘Matric Groups’, Amer. Math. Monthly 67 (1960) 268.

    Article  Google Scholar 

  • Wallace, A. D., ‘Remarks on Affine Semigroups’, Bull. Amer. Math. Soc. 66 (1960) 110–112.

    Article  Google Scholar 

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© 1974 D. Reidel Publishing Company, Dordrecht-Holland

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Mulcrone, T. (1974). Remarks on Two-By-Two Matric Semigroups. In: Cohen, R.S., Stachel, J.J., Wartofsky, M.W. (eds) For Dirk Struik. Boston Studies in the Philosophy of Science, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2115-9_8

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  • DOI: https://doi.org/10.1007/978-94-010-2115-9_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-277-0379-8

  • Online ISBN: 978-94-010-2115-9

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