Abstract
Š. Schwarz in his paper [SCHWARZ, Š.: Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.: Contribution to the theory of torsion semigroups, Czechoslovak Math. J. 3(1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.
The author was supported by the Grant Agency of the Czech Republic, Grant # 17-02804S and the strategic development financing RVO 67985807. All computations are made using Mathematica 10 program package.
Dedicated to the memory of Professor Štefan Schwarz
(Communicated by Milan Paštéka)
References
[1] Abrhan, I.: On prime ideals in groupoids and in multiplicative semigroups of residue classes (mod m), Math. Slovaca 34 (1984), 121–133 (in Russian).Search in Google Scholar
[2] Alahmadi, A.—Jain, S. K.—Leroy, A.: Decomposition of singular matrices into idempotents, Linear Multilinear Algebra 62(1) (2014), 13–27.10.1080/03081087.2012.754439Search in Google Scholar
[3] Alomair, B.—Clark, A.—Poovendran, R.: The power of primes: security of authentication based on a universal hash-function family, J. Math. Cryptol. 4(2) (2010), 121–148.10.1515/jmc.2010.005Search in Google Scholar
[4] Arbib, M. A. (ed.): Algebraic Theory of Machines, Languages, and Semigroups, Academic Press, New York and London, 1968.Search in Google Scholar
[5] Autonne, L.: Sur certains groupes linéaires, C. R. Math. Acad. Sci. Paris 143 (1907), 670–672.Search in Google Scholar
[6] Autonne, L.: Sur les groupes de matrices linéaires non invertibles, Ann. Univer. Lyon (NS) (1909), Fasc. 25, 78 pp.Search in Google Scholar
[7] Bachmann, P.: Niedere Zahlentheorie, Vol. 1, B. G. Teubner, Leipzig, 1902.Search in Google Scholar
[8] Ballantine, C. S.: Products of idempotent matrices, Linear Algebra Appl. 19 (1978), 81–86.10.1016/0024-3795(78)90006-XSearch in Google Scholar
[9] Bauer, M.: Sur les congruences identiques, Nouv. Ann. Math. (4) 2 (1902), 256–264.Search in Google Scholar
[10] Bauer, M.: Lösung zu 282, (P. Muth.), Arch. der Math. u. Phys., 1910.Search in Google Scholar
[11] Bell, E. T.: Unique decomposition, Amer. Math. Monthly 37(8) (1930), 400–418.10.1080/00029890.1930.11987103Search in Google Scholar
[12] Bell, E. T.: Finite ova, Proc. Natl. Acad. Sci. USA 19 (1933), 577–579.10.1073/pnas.19.5.577Search in Google Scholar PubMed PubMed Central
[13] Bergelson, V.: Minimal idempotents and ergodic Ramsey theory. In: Topics in Dynamics and Ergodic Theory. Survey papers and mini-courses presented at the international conference and US-Ukrainian workshop on dynamical systems and ergodic theory, Katsiveli, Ukraine, August 21–30, 2000, London Math. Soc. Lecture Note Ser. 310, Cambridge University Press, Cambridge, 2003, pp. 8–39.10.1017/CBO9780511546716.004Search in Google Scholar
[14] Berman, A.—Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences. Computer Science and Applied Mathematics, Academic Press, New York, San Francisco, London, 1979.10.1016/B978-0-12-092250-5.50009-6Search in Google Scholar
[15] Berthé, V.—Rigo, M. (eds.): Combinatorics, Words and Symbolic Dynamics. Encyclopedia of Mathematics and its Applications, Vol. 159, Cambridge Univ. Press, Cambridge, 2016.10.1017/CBO9781139924733Search in Google Scholar
[16] Bhaskara Rao, K. P. S.: Products of idempotent matrices over integral domains, Linear Algebra Appl. 430(10) (2009), 2690–2695.10.1016/j.laa.2008.11.018Search in Google Scholar
[17] Bhaskara Rao, K. P. S.: Products and sums of idempotent matrices over principal ideal domains. In: Combinatorial Matrix Theory and Generalized Inverses of Matrices (New Delhi), Springer, 2013, pp. 171–175.10.1007/978-81-322-1053-5_14Search in Google Scholar
[18] Blizard, W. D.: The development of multiset theory, Mod. Log. 1(4) (1991), 319–352.Search in Google Scholar
[19] Blizard, W. D.: Updates and corrections to “The development of multiset theory”, Mod. Log. 7(3–4) (1997), 434.Search in Google Scholar
[20] Bobok, J.—Snoha, L.: Periodic points and little Fermat theorem, Nieuw Arch. Wiskd. (4) 10(1–2) (1992), 33–35.Search in Google Scholar
[21] Boole, G.: The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning, Macmillan, Barclay, Cambridge, London, 1847.Search in Google Scholar
[22] Butler, K. K-H.—Krabill, J. R.: Circulant Boolean relation matrices, Czechoslovak Math. J. 24 (1974), 247–251.10.21136/CMJ.1974.101235Search in Google Scholar
[23] Cao, Y.: Multiplicative semigroup of a residue class ring, Semigroup Forum 70(3) (2005), 361–368.10.1007/s00233-004-0144-ySearch in Google Scholar
[24] Cayley, A.: A memoir on the theory of matrices, Philos. Trans Roy. Soc. London 148 (1858), 17–37.10.1017/CBO9780511703683.053Search in Google Scholar
[25] Cayley, A.: On multiple algebra, Q. J. Pure Appl. Math. XXII (1887), 270–308; Coll. Math. Papers, Vol. 12, 1897.Search in Google Scholar
[26] Cayley, A.: Presidential address to the British association, Southport, september 1883. In: Coll. Math. Papers, Vol. 11, Cambridge University Press, 2009, pp. 429–459.10.1017/CBO9780511703775.080Search in Google Scholar
[27] Chao, Ch.-Y.—Zhang, M. Ch.: On generalized circulants over a Boolean algebra, Linear Algebra Appl. 62 (1984), 195–206.10.1016/0024-3795(84)90095-8Search in Google Scholar
[28] Clifford, A. H.: Semigroups admitting relative inverses, Ann. Math. 42(4) (1941), 1037–1049.10.2307/1968781Search in Google Scholar
[29] Clifford, A. H.—Preston, G. B.: The Algebraic Theory of Semigroups. Math. Surveys Monogr. 7, Amer. Math. Soc., Providence, R.I., 1961.10.1090/surv/007.1Search in Google Scholar
[30] Conway, A. W.: Quaternions and matrices, Proc. R. Ir. Acad., Sect. A 50 (1945), 98–103.Search in Google Scholar
[31] Coons, M. J.: The emergence of modern number theory in Hungary. In: Fulbright Student Conference Papers II, Hungarian-American Commission for Educational Exchange (Published by the Fulbright Commission), Budapest, 2009, pp. 181–192.Search in Google Scholar
[32] Cross, J. T. The Euler phi-function in the Gaussian integers, Amer. Math. Monthly 90 (1983), 518–528.10.1080/00029890.1983.11971275Search in Google Scholar
[33] Dedekind, R.: Essays on the Theory of Numbers, I. Continuity and Irrational Numbers, II. The Nature and Meaning of Numbers, authorized translation by W. W. Beman, The Open Court Publishing Co., Chicago and Kegan Paul and Co., London, 1901.Search in Google Scholar
[34] Dénes, J.—Hermann, P.: On the product of all elements in a finite group, Ann. Discrete Math. 15 (1982), 105–109.10.1016/S0304-0208(08)73257-2Search in Google Scholar
[35] Dickson, L. E.: On semi-groups and the general isomorphism between finite groups, Trans. Amer. Math. Soc. 6(2) (1905), 205–208.10.1090/S0002-9947-1905-1500707-4Search in Google Scholar
[36] Dickson, L. E.: History of the Theory of Numbers, Vol. I, Carnegie Institution of Washington, New York, 1919.10.5962/t.174869Search in Google Scholar
[37] Dickson, L. E.: Algebras and Their Arithmetics, Univ. of Chicago Press, Chicago, 1923.Search in Google Scholar
[38] Dickson, L. E.: Éléments de la Théorie des Groupes Abstraits, by J.-A. de Séguier, Bull. Amer. Math. Soc. 11(3) (1904), 159–162.10.1090/S0002-9904-1904-01201-XSearch in Google Scholar
[39] Dieudonné, J.: Introductory remarks on algebra, topology and analysis, Hist. Math. 2 (1975), 537–548.10.1016/0315-0860(75)90118-4Search in Google Scholar
[40] Dulmage, A. L.—Mendelsohn, N. S.: Graphs and matrices. In: Graph Theory Theoretical Physics, F. Harary (ed.), Academic Press, 1967, pp. 167–227.Search in Google Scholar
[41] Ecker, A.: Finite semigroups and the RSA-cryptosystem. Cryptography, Proc. Workshop, Burg Feuerstein/Ger. 1982, In: Lect. Notes Comput. Sci. 149, 1983, pp. 353–369.10.1007/3-540-39466-4_26Search in Google Scholar
[42] Ellis, R.: Lectures on Topological Dynamics, W. A. Benjamin, Inc., New York, 1969.Search in Google Scholar
[43] Erdős, P.—Turán, P.: On some problems of a statistical group-theory. I., Z. Wahrscheinlichkeitstheor. Verw. Geb. 4 (1965), 175–186.10.1007/BF00536750Search in Google Scholar
[44] Erdős, P.—Turán, P.: On some problems of a statistical group-theory. III., Acta Math. Acad. Sci. Hung. 18 (1967), 309–320.10.1007/BF02280290Search in Google Scholar
[45] Erdos, J. A.: On products of idempotent matrices, Glasg. Math. J. 8 (1967), 118–122.10.1017/S0017089500000173Search in Google Scholar
[46] Eschenbach, C.: Idempotence for sign-pattern matrices, Linear Algebra Appl. 180 (1993), 153–165.10.1016/0024-3795(93)90529-WSearch in Google Scholar
[47] Farahat, H. K.—Mirsky, L.: A condition for diagonability of matrices, Amer. Math. Monthly 63(6) (1956), 410–412.10.2307/2309405Search in Google Scholar
[48] Farahat, H. K.—Mirsky, L.: Group membership in rings of various types, Math. Z. 70 (1958), 231–244.10.1007/BF01558587Search in Google Scholar
[49] Farebrother, R. W.—Gross, J.—Troschke, S.-O.: Matrix representation of quaternions, Linear Algebra Appl. 362 (2003), 251–255.10.1016/S0024-3795(02)00535-9Search in Google Scholar
[50] Feinberg, R. B.: Similarity of partitioned matrices, J. Res. Natl. Bur. Stand., Sect. B 79 (1976), 117–125.10.6028/jres.079B.012Search in Google Scholar
[51] Feit, W.: A group-theoretic proof of Wilson’s Theorem, Amer. Math. Monthly 65 (1958), 120.10.2307/2308892Search in Google Scholar
[52] Flachsmeyer, J.—Katrnoška, F.: On the number of the idempotents of some matrix rings, Tatra Mt. Math. Publ. 10 (1997), 129–132.Search in Google Scholar
[53] Frobenius, G.: Ueber lineare Substitutionen und bilineare Formen, J. Reine Angew. Math. LXXXIV (1877), 1–63.10.1515/9783112335789-002Search in Google Scholar
[54] Frobenius, G.: Neuer Beweis des Sylow’schen Satzes, J. Reine Angew. Math. C (1886), 179–181.10.1515/crll.1887.100.179Search in Google Scholar
[55] Frobenius, G.: Über endliche Gruppen, Sitzungsber. Königl. Preuß Akad. Wiss. Berlin (1895), 163–194.Search in Google Scholar
[56] Frobenius, G.: Über Matrizen aus nicht negativen Elementen, Sitzungsber. Königl. Preuß. Akad. Wiss. Berlin (1912), 456–477.Search in Google Scholar
[57] Frobenius, G.—Schur, I.: Über die Äquivalenz der Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. Wiss. Berlin (1906), 209–217.Search in Google Scholar
[58] Furstenberg, H.—Katznelson, Y.: Idempotents in compact semigroups and Ramsey theory, Israel J. Math. 68(3) (1989), 257-270.10.1007/BF02764984Search in Google Scholar
[59] Gantmacher, F. R.: The Theory of Matrices, Vol. 1 & 2, AMS Chelsea Publishing, Providence, R.I., 1998.Search in Google Scholar
[60] Ganyushkin, O.—Mazorchuk, V.: Classical Finite Transformation Semigroups. An Introduction, Springer, London, 2009.10.1007/978-1-84800-281-4Search in Google Scholar
[61] Gauss, C. F.: Disquisitiones Arithmeticae, (transl. from the Latin by A. A. Clarke, rev. by W. C. Waterhouse, with the help of C. Greither and A. W. Grootendorst), reprint of the 1966 ed., Springer-Verlag, New York etc., 1986.10.1007/978-1-4939-7560-0Search in Google Scholar
[62] Gillam, D. W. H.—Hall, T. E.—Williams, N. H.: On finite semigroups and idempotents, Bull. Lond. Math. Soc. 4 (1972), 143–144.10.1112/blms/4.2.143Search in Google Scholar
[63] Goldstein, C.—Schappacher, N.—Schwermer, J. (eds.): The Shaping of Arithmetic After C. F. Gauss’s Disquisitiones Arithmeticae, Springer, Berlin, 2007.10.1007/978-3-540-34720-0Search in Google Scholar
[64] Grabmeier, J.—Kaltofen, E.—Weispfenning, V. (eds.): Computer Algebra Handbook. Foundations, Applications, Systems, Springer, Berlin, 2003.10.1007/978-3-642-55826-9Search in Google Scholar
[65] Graham, R. L.: On finite o-simple semigroups and graph theory, Math. Systems Theory 2 (1968), 325–339.10.1007/BF01703263Search in Google Scholar
[66] Graham, N.—Graham, R.—Rhodes, J.: Maximal subsemigroups of finite semigroups, J. Comb. Theory 4 (1968), 203–209.10.1016/S0021-9800(68)80001-8Search in Google Scholar
[67] Grošek, O.: A topology on semigroups generated by an endomorphism, Math. Slovaca 28 (1978), 217–223.Search in Google Scholar
[68] Grošek, O.: Remarks concerning RSA-cryptosystem exponents, Math. Slovaca 44(2) (1994), 279–285.Search in Google Scholar
[69] Grošek, O.: On a reconstruction of a Markov chain, J. Combin. Inform. System Sci. 20(1–4) (1995), 85–93.Search in Google Scholar
[70] Grošek, O.—Nemoga, K.—Satko, L.: Remarks to short RSA public exponents. Number theory (Cieszyn, 1998), Ann. Math. Sil. 12 (1998), 65–74.Search in Google Scholar
[71] Grošek, O.—Porubský, Š.: Coprime solutions to ax ≡ b (mod n), J. Math. Cryptol. 7 (2013), 217–224.10.1515/jmc-2013-5003Search in Google Scholar
[72] Grošek, O.—ŠIška, J.: Semigroup of matrices over GF(2S) and its relation to AES, Comput. Inform. 22(5) (2003), 417–426.Search in Google Scholar
[73] Grošek, O.—ŠIška, J.: Signature schemes based on matrices, Congr. Numer. 156 (2002), 69–80.Search in Google Scholar
[74] Grošek, O.—Vojvoda, M.—Krchnavý, R.: A new matrix test for randomness, Computing 85(1–2) (2009), 21–36.10.1007/s00607-009-0033-zSearch in Google Scholar
[75] Grošek, O.—Zajac, P.: Efficient selection of the AES-class MixColumns parameters. In: WSEAS Transactions on Information Science and Applications, Vol. 4, 2007, pp. 663–668.Search in Google Scholar
[76] Gross, J.: Idempotency of the Hermitian part of a complex matrix, Linear Algebra Appl. 289(1–3) (1999), 135–139.10.1016/S0024-3795(97)10045-3Search in Google Scholar
[77] Gruber, F.: Zur Theorie der Fermat’schen Congruenzen, Mathem. und Naturwiss. Berichte aus Ungarn 13 (1896), 413–417.Search in Google Scholar
[78] Guralnick, R. M.—Lorenz, M.: Orders of finite groups of matrices. In: Groups, Rings and Algebras, A conference in honor of Donald S. Passman, Madison, WI, USA, June 10–12, 2005, Amer. Math. Soc., Providence, R.I, 2006, pp. 141–161.10.1090/conm/420/07974Search in Google Scholar
[79] Guričan, J.: The semigroup of general circulant matrices, Acta Math. Univ. Comenian. 44–45 (1984), 13–21.Search in Google Scholar
[80] Gustafson, W. H.: The history of algebras and their representations. In: Representations of Algebras (M. Auslander and E. Lluis (eds.)), Lecture Notes in Math. 944, Springer, Berlin, Heidelberg, 1982, pp. 1–28.Search in Google Scholar
[81] Hall, T. E.—Sapir, M. V.: Idempotents, regular elements and sequences from finite semigroups, Discrete Math. 161(1–3) (1996), 151–160.10.1016/0012-365X(95)00223-JSearch in Google Scholar
[82] Hardy, G. H.—Wright, E. M.: An Introduction to the Theory of Numbers, 5th ed., Oxford at the Clarendon Press., Oxford etc., 1979.Search in Google Scholar
[83] Hardy, G. H.—Wright, E. M.: Leudesdorf’s extension of Wolstenholme’s theorem, J. Lond. Math. Soc. 9 (1934), 38–41.10.1112/jlms/s1-9.1.38Search in Google Scholar
[84] Hardy, G. H.—Wright, E. M.: Leudesdorf’s extension of Wolstenholme’s theorem. Corrigendum, J. Lond. Math. Soc. 9 (1934), 240.10.1112/jlms/s1-9.3.240-sSearch in Google Scholar
[85] Hermann, P. Z.: On the product of all nonzero elements of a finite ring, Glasg. Math. J. 30(3) (1968), 325–330.10.1017/S0017089500007412Search in Google Scholar
[86] Hermann, P. Z.—Robertson, E. F.—Ruškuc, N.: On products of all elements of a finite semigroup, Proc. Edinb. Math. Soc. (2) 42(3) (1999), 551–557.10.1017/S0013091500020514Search in Google Scholar
[87] Hewitt, E.—Zuckerman, H. S.: The multiplicative semigroup of integers modulo m, Pacific J. Math. 10 (1960), 1291–1308.10.2140/pjm.1960.10.1291Search in Google Scholar
[88] Hilton, H.: An Introduction to the Theory of Groups of Finite Order, Clarendon Press, Oxford, 1908.Search in Google Scholar
[89] Hindman, N.—Strauss, D.: Algebra in the Stone-čech Compactification: Theory and Applications, Walter de Gruyter, Berlin, 1998.10.1515/9783110809220Search in Google Scholar
[90] Hollings, C.: Mathematics Across the Iron Curtain. A History of the Algebraic Theory of Semigroups. History of Mathematics, Vol. 41., Amer. Math. Soc. Providence, R.I., 2014.10.1090/hmath/041Search in Google Scholar
[91] Huntington, E. V.: Simplified definition of a group, Bull. Amer. Math. Soc. 2(8) (1902), 296–300.10.1090/S0002-9904-1902-00898-7Search in Google Scholar
[92] Isaacson, D.—Madsen, R.: Positive columns for stochastic matrices, J. Appl. Probab. 11 (1974), 829–835.10.2307/3212567Search in Google Scholar
[93] Jacobson, N.: Structure of Rings. Amer. Math. Soc. Colloq. Publ. 37, Amer. Math. Soc., Providence, R.I., 1956.10.1090/coll/037Search in Google Scholar
[94] Kaplansky, I.: A theorem on division rings, Canad. J. Math. 3 (1951), 290–292.10.1007/978-1-4612-5352-5_9Search in Google Scholar
[95] Khatri, C. G.: A note on idempotent matrices, Linear Algebra Appl. 70 (1985), 185–195.10.1016/0024-3795(85)90051-5Search in Google Scholar
[96] Khatri, C. G.: Powers of matrices and idempotency, Linear Algebra Appl. 33 (1980), 57–65.10.1016/0024-3795(80)90097-XSearch in Google Scholar
[97] Kim, J. B.: On circulant fuzzy matrices, Math. Japon. 24(1) (1979/80), 35–40.Search in Google Scholar
[98] Kim, K. H.: Boolean Matrix Theory and Applications. Pure Appl. Math. 70, Marcel Dekker, Inc, New York, Basel, 1982.Search in Google Scholar
[99] Kimura, N.: Maximal subgroups of a semigroup, Kōdai Math. Semin. Rep. (1954), 85–88.10.2996/kmj/1138843500Search in Google Scholar
[100] Klein, A. A.: On Fermat’s theorem for matrices and the periodic identities of Mn(GF(q)), Arch. Math. 34 (1980), 399–402.10.1007/BF01224977Search in Google Scholar
[101] Koch, R. J.: Remarks on primitive idempotents in compact semigroups with zero, Proc. Amer. Math. Soc. 5 (1954), 828–833.10.1090/S0002-9939-1954-0065569-XSearch in Google Scholar
[102] Kolibiarová, B.: Partially commutative torsion semigroups, Mat.-Fyz. Čas. Slovensk. Akad. Vied 9(3) (1959), 160–170 (in Slovak).Search in Google Scholar
[103] Kollár, D.—Šulka, R.: On some closure operators on semigroups, Demonstr. Math. 24(1–2) (1991), 35–45.10.1515/dema-1991-1-206Search in Google Scholar
[104] Konieczny, J.: Remarks on the structure of the multiplicative monoid of integers modulo m, Semigroup Forum 46(2) (1993), 266–269.10.1007/BF02573569Search in Google Scholar
[105] Kou, L. T.: Fixed length sources and sinks of a graph, Math. Jap. 29 (1984), 411–417.Search in Google Scholar
[106] Kuzmanovich, J.—Pavlichenkov, A.: Finite groups of matrices whose entries are integers, Amer. Math. Monthly 109(2) (2002), 173–186.10.1080/00029890.2002.11919850Search in Google Scholar
[107] Laduke, J.: The study of linear associative algebras in the United States, 1870–1927. In: Emmy Noether in Bryn Mawr (B. Srinivasan and J. D. Sally, eds.), Springer, New York, 1983, pp. 147–159.10.1007/978-1-4612-5547-5_11Search in Google Scholar
[108] Lam, T. Y.: Corner ring theory: a generalization of Peirce decompositions, I. In: Algebras, Rings and Their Representations. Proceedings of the international conference on algebras, modules and rings, University of Lisbon, Lisbon, Portugal, July 14–18, 2003, World Scientific, Hackensack, N.J., 2006, pp. 153–182.10.1142/9789812774552_0011Search in Google Scholar
[109] Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, B. G. Teubner, Leipzig &. Berlin, 1909.Search in Google Scholar
[110] Laššák, M.: Bauer’s congruence in algebraic number fields, Tatra Mt. Math. Publ. 20 (2000), 143–154.Search in Google Scholar
[111] Laššák, M.: Wilson’s theorem in algebraic number fields, Math. Slovaca 50(3) (2000), 303–314.Search in Google Scholar
[112] Laššák, M.—Porubský, Š.: Fermat-Euler theorem in algebraic number fields, J. Number Theory 60(2) (1996), 254–290.10.1006/jnth.1996.0123Search in Google Scholar
[113] Ljapin, E. S.: Semigroups, Transl. Math. Monogr. 3., Amer. Math. Soc., Providence, R.I., 1963.10.1090/mmono/003Search in Google Scholar
[114] Ljubič, Ju. I.: Estimates for the optimal determination of indeterminate autonomous automata, Sibirsk. Mat. Ž. 5 (1964), 337–355.Search in Google Scholar
[115] Lombardi, H.—Quitté, C.: Commutative Algebra: Constructive Methods. Finite Projective Modules. Algebr. Appl. 20, Springer, Dordrecht Heidelberg New York London, 2015.10.1007/978-94-017-9944-7Search in Google Scholar
[116] Lubelski, S.: Zur Theorie der höheren Kongruenzen, J. Reine Angew. Math. 162 (1930), 63–68.10.1515/crll.1930.162.63Search in Google Scholar
[117] Mackiw, G.: Finite groups of 2× 2 integer matrices, Math. Mag. 69(5) (1996), 356–361.10.1080/0025570X.1996.11996474Search in Google Scholar
[118] Marshall, J. B.: On the extension of Fermat’s theorem to matrices of order n, Proc. Edinb. Math. Soc. (2) 6 (1939), 85–91.10.1017/S0013091500024536Search in Google Scholar
[119] Mazur, M.—Petrenko, B. V.: Generalizations of Arnold’s version of Euler’s theorem for matrices, Jpn. J. Math. 5(2) (2010), 183–189.10.1007/s11537-010-1023-9Search in Google Scholar
[120] Mazur, M.—Petrenko, B. V.: Addendum to: “Generalizations of Arnold’s version of Euler’s theorem for matrices”, Jpn. J. Math. 6(1) (2011), 63–64.10.1007/s11537-011-1112-4Search in Google Scholar
[121] McKenzie, R.—Schein, B. M.: Every semigroup is isomorphic to a transitive semigroup of binary relations, Trans. Amer. Math. Soc. 349 (1077), 271–285.10.1090/S0002-9947-97-01708-XSearch in Google Scholar
[122] Mehra, J.—Rechenberg, H.: The Historical Development of Quatum Theory, Vol. 3 (The Formulation of Matrix Mechanics and its Modifications 1925–1926), Springer Verlag, New York, 1982.10.1007/978-1-4612-5781-3_1Search in Google Scholar
[123] Meyer, C. D., Jr.: The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev. 17 (1975), 443–464.10.1137/1017044Search in Google Scholar
[124] Meyer, C. D., Jr.: Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia (PA), 2000.10.1137/1.9780898719512Search in Google Scholar
[125] Miller, G. A.: A new proof of the generalized Wilson’s theorem, Ann. Math. 4(4) (1903), 188–190.10.2307/1967333Search in Google Scholar
[126] Moore, E. H.: A definition of abstract groups, Trans. Amer. Math. Soc. 3(4) (1902), 485–492.10.1090/S0002-9947-1902-1500616-8Search in Google Scholar
[127] Moore, E. H.: Notes and errata: ‘A definition of abstract groups’, Trans. Amer. Math. Soc. 3 (1902), no. 4, 485–492, Trans. Amer. Math. Soc. 5(4) (1904), 549.10.1090/S0002-9947-1902-1500616-8Search in Google Scholar
[128] Moore, E. H.: On a definition of abstract groups, Trans. Amer. Math. Soc. 6(2) (1905), 179–180.10.1090/S0002-9947-1905-1500704-9Search in Google Scholar
[129] Munn, W. D.: Pseudo-inverses in semigroups, Proc. Camb. Philos. Soc. 57(2) (1961), 247–250.10.1017/S0305004100035143Search in Google Scholar
[130] Newman, M.: Integral Matrices. Pure Appl. Math. 45, Academic Press, New York, London, 1972.Search in Google Scholar
[131] Niven, I.: The roots of a quaternion, Amer. Math. Monthly 49 (1942), 386–388.10.1080/00029890.1942.11991248Search in Google Scholar
[132] Niven, I.: Fermat’s theorem for matrices, Duke Math. J. 15 (1948), 823–826.10.1215/S0012-7094-48-01574-9Search in Google Scholar
[133] Numakura, K.: On bicompact semigroups, Math. J. Okayama Univ. 1 (1952), 99–108.Search in Google Scholar
[134] Olshevsky, V.: Similarity of block diagonal and block triangular matrices, Integral Equations Operator Theory 15 (1992), 853–863.10.1007/BF01200704Search in Google Scholar
[135] Parker, E. T.: On multiplicative semigroups of residue classes, Proc. Amer. Math. Soc. 5 (1954), 612–616.10.1090/S0002-9939-1954-0062732-9Search in Google Scholar
[136] Parízek, B.: On the decomposition of the semigroup of residue classes (mod m) into a direct product, Mat.-Fyz. Časopis Slovensk. Akad. Vied 10 (1960), 18–27 (in Slovak).Search in Google Scholar
[137] Parízek, B.: Note on structure of multiplicative semigroup of residue classes, Mat.-Fyz. Časopis Slovensk. Akad. Vied 7(3) (1957), 183–185 (in Slovak).Search in Google Scholar
[138] Parízek, B.—Schwarz, Š.: On the multiplicative semigroup of residue classes (mod m), Mat.-Fyz. Časopis Slovensk. Akad. Vied 8(3) (1958), 136–150 (in Slovak).Search in Google Scholar
[139] Paz, A.: Introduction to Probabilistic Automata. Computer Science and Applied Mathematics, Academic Press, New York, London, 1971.Search in Google Scholar
[140] Peirce, B.: Linear Associative Algebra, Lithograph, Washington, D.C., 1870. (http://www.math.harvard.edu/history/peirce_algebra/index.html)Search in Google Scholar
[141] Peirce, B.: Linear associative algebra, Amer. J. Math. 4(1) (1881), 97–229.10.2307/2369153Search in Google Scholar
[142] Peirce, B.: Linear Associative Algebra, new ed. with Addenda and Notes, by C.S.Peirce, Son of the Author, van Nostrand, New York, 1882.Search in Google Scholar
[143] Peirce, B.: On the uses and transformations of linear algebra, Proc. Amer. Acad. Arts Sciences 10 (1875), 395–400.10.2307/20021428Search in Google Scholar
[144] Poole, A. B.: Finite ova, Amer. J. Math. 59 (1937), 23–32.10.2307/2371556Search in Google Scholar
[145] Porubský, Š.: Idempotents and congruence ax ≡ b (mod n). In: “From Arithmetic to Zeta-Functions. Number Theory in Memory of Wolfgang Schwarz” (J. Sander, J. Steuding, R. Steuding, eds.), Springer Verlag, Cham, 2016, pp. 385–403.10.1007/978-3-319-28203-9_23Search in Google Scholar
[146] Porubský, Š.: Semigroup structure of sets of solutions to equation Xm=Xs, Ann. Univ. Sci. Budapest., Sect. Comput. 48 (2018), 151–167.Search in Google Scholar
[147] Pohst, M.—Zassenhaus, H.: Algorithmic Algebraic Number Theory. Encyclopedia of Mathematics and its Applications 30, Cambridge University Press, Cambridge etc., 1989.10.1017/CBO9780511661952Search in Google Scholar
[148] Prosvirov, A. S.: On periodic semigroups, Mat. Zap. Sverdl. 8(1) (1971), 77–94.Search in Google Scholar
[149] Pujol, J.: Hamilton, Rodrigues, Gauss, quaternions, and rotations: a historical reassessment, Commun. Math. Anal. 13(2) (2012), 1–14.Search in Google Scholar
[150] Ranum, A.: Concerning linear substitutions of finite period with rational coefficients, Trans. Amer. Math. Soc. 9(2) (1908), 183–202.10.1090/S0002-9947-1908-1500808-3Search in Google Scholar
[151] Ranum, A.: On the matrices of period a power of p in Jordan’s linear congruence groups, modulo p2, Bull. Amer. Math. Soc. 13(7) (1907), 336–345.10.1090/S0002-9904-1907-01476-3Search in Google Scholar
[152] Ranum, A.: On periodic linear substitutions whose coefficients are integers, Bull. Amer. Math. Soc. 15(1) (1908), 4–6.10.1090/S0002-9904-1908-01683-5Search in Google Scholar
[153] Ranum, A.: The number of classes of conjugate periodic linear substitutions with rational coefficients, Jahresber. Dtsch. Math.-Ver. 17 (1908), 234–236.Search in Google Scholar
[154] Ranum, A.: The group-membership of singular matrices, Amer. J. Math. 31 (1909), 18–41.10.2307/2370176Search in Google Scholar
[155] Ranum, A.: The group of classes of congruent quadratic integers with respect to a composite ideal modulus, Amer. Math. Soc. Trans. 11 (1910), 172–198.10.1090/S0002-9947-1910-1500859-8Search in Google Scholar
[156] Ranum, A.: The groups belonging to a linear associative algebra, Amer. J. Math. 49 (1927), 285–308.10.2307/2370757Search in Google Scholar
[157] Richardson, L.—Ruby, S.: RESTful Web Services, O’Reilly Media, 2007.Search in Google Scholar
[158] Riguet, J.: Relations binaires, fermetures, correspondances de Galois, Bull. Soc. Math. Fr. 76 (1948), 114–154.10.24033/bsmf.1401Search in Google Scholar
[159] Rowen, L. H.: Ring Theory, Vol. I, Academic Press, Inc., Boston (MA), 1988.Search in Google Scholar
[160] Rystsov, I. K.: Primitive and irreducible automata, Cybernet. Systems Anal. 51(4) (2015), 506–513 (translation of Kibernet. Sistem. Anal. 2(4) (2015), 19–27).10.1007/s10559-015-9742-9Search in Google Scholar
[161] Schafer, R. D.: An Introduction to Nonassociative Algebras. Pure Appl. Math. 22, A Series of Monographs and Textbooks, Academic Press, New York and London, 1966.Search in Google Scholar
[162] Schein, B. M.: Relation algebras and function semigroups, Semigroup Forum 1(1) (1970), 1–62.10.1007/BF02573019Search in Google Scholar
[163] Schein, B. M.: On certain classes of semigroups of binary relations, Sibirsk. Mat. Ž. 6 (1965), 616–635 [Amer. Math. Soc. Transl. Ser. 2, Vol. 139, Amer. Math. Soc., Providence, R.I., 1988].Search in Google Scholar
[164] Schur, I.: Neue Begründung der Theorie der Gruppencharaktere, Sitzungsber. Preuss. Akad. Wiss. Berlin (1904, 1905), 406–432.10.1007/978-3-642-61947-2_7Search in Google Scholar
[165] Schwarz, Š.: Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp. (in Slovak).Search in Google Scholar
[166] Schwarz, Š.: On various generalization of the notion of a group, Časopis Pěst. Mat. Fys. 74(2) (1949), 95–113 (in Slovak).10.21136/CPMF.1949.123053Search in Google Scholar
[167] Schwarz, Š.: Contribution to the theory of torsion semigroups, Czechoslovak Math. J. 3(1) (1953), 7–21 (in Russian).10.21136/CMJ.1953.100067Search in Google Scholar
[168] Schwarz, Š.: On a Galois connexion in the theory of characters of commutative semigroups, Czechoslovak Math. J. 4 (1954), 296–313.10.21136/CMJ.1954.100118Search in Google Scholar
[169] Schwarz, Š.: On Hausdorff bicompact semigroups, Czechoslovak Math. J. 5(1) (1955), 1–13 (in Russian).10.21136/CMJ.1955.100128Search in Google Scholar
[170] Schwarz, Š.: Semigroups satisfying some weakened forms of the cancellation law, Mat.-Fyz. Časopis Slovensk. Akad. Vied 6(3) (1956), 149–158 (in Slovak).Search in Google Scholar
[171] Schwarz, Š.: An elementary semigroup theorem and a congruence relation of Rédei, Acta Sci. Math. Szeged 19 (1958), 1–4.Search in Google Scholar
[172] Schwarz, Š.: Remark on the theory of non-negative matrices, Sibirsk. Mat. Ž. 6 (1965), 207–211 (in Russian).Search in Google Scholar
[173] Schwarz, Š.: A semigroup treatment of some theorems on non-negative matrices, Czechoslovak Math. J. 15(2) (1965), 212–229.10.21136/CMJ.1965.100663Search in Google Scholar
[174] Schwarz, Š.: On the structure of the semigroup of stochastic matrices, Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1965), 297–311.Search in Google Scholar
[175] Schwarz, Š.: On powers of non-negative matrices, Mat.-Fyz. Časopis Slovensk. Akad. Vied 15(3) (1965), 215–228.Search in Google Scholar
[176] Schwarz, Š.: New kinds of theorems on non-negative matrices, Czechoslovak Math. J. 16(2) (1966), 285–295.10.21136/CMJ.1966.100730Search in Google Scholar
[177] Schwarz, Š.: On the index of imprimitivity of a non-negative matrix, Acta Sci. Math. (Szeged) 28 (1967), 185–189.Search in Google Scholar
[178] Schwarz, Š.: A note on the structure of the semigroup of doubly-stochastic matrices, Mat. Časopis Slovensk. Akad. Vied 17 (1967), 308–316.Search in Google Scholar
[179] Schwarz, Š.: Some estimates in the theory of non-negative matrices, Czechoslovak Math. J. 17 (1967), 399–407.10.21136/CMJ.1967.100785Search in Google Scholar
[180] Schwarz, Š.: L’application des demi-groupes à l’étude des matrices non-négatives, Sem. P. Dubreil, Algebre Theorie Nombres 20(2) (1966/67), (1968), 8 pp.Search in Google Scholar
[181] Schwarz, Š.: On a sharp estimation in the theory of binary relations on a finite set, Czechoslovak Math. J. 20(4) (1970), 703–714.10.21136/CMJ.1970.100992Search in Google Scholar
[182] Schwarz, Š.: On idempotent binary relations on a finite set, Czechoslovak Math. J. 20(4) (1970), 696–702.10.21136/CMJ.1970.100991Search in Google Scholar
[183] Schwarz, Š.: On the semigroup of binary relations on a finite set, Czechoslovak Math. J. 20(4) (1970), 632–679.10.21136/CMJ.1970.100989Search in Google Scholar
[184] Schwarz, Š.: The semigroup of fully indecomposable relations and Hall relations, Czechoslovak Math. J. 23 (1973), 151–163.10.21136/CMJ.1973.101153Search in Google Scholar
[185] Schwarz, Š.: Circulant Boolean relation matrices, Czechoslovak Math. J. 24(2) (1974), 252–253.10.21136/CMJ.1974.101236Search in Google Scholar
[186] Schwarz, Š.: Sums of powers of binary relations, Mat. Časopis Slovensk. Akad. Vied 24 (1974), 161–171.Search in Google Scholar
[187] Schwarz, Š.: A counting theorem in the semigroup of circulant Boolean matrices, Czechoslovak Math. J. 27 (1977), 504–510.10.21136/CMJ.1977.101485Search in Google Scholar
[188] Schwarz, Š.: The Euler-Fermat theorem for the semigroup of circulant Boolean matrices, Czechoslovak Math. J. 30(1) (1980), 135–141.10.21136/CMJ.1980.101663Search in Google Scholar
[189] Schwarz, Š.: The role of semigroups in the elementary theory of numbers, Math. Slovaca 31(4) (1981), 369–395.10.21136/CMJ.1981.101733Search in Google Scholar
[190] Schwarz, Š.: An unconventional problem in the elementary theory of numbers, Czechoslovak Math. J. 31(1) (1981), 159–169.10.21136/CMJ.1981.101733Search in Google Scholar
[191] Schwarz, Š.: Extensions of Bauer’s identical congruences, Math. Slovaca 33(2) (1983), 209–224.Search in Google Scholar
[192] Schwarz, Š.: Fermat’s theorem for matrices revisited, Math. Slovaca 35(4) (1985), 343–347.Search in Google Scholar
[193] Schwarz, Š.: Common consequents in directed graphs, Czechoslovak Math. J. 35 (1985), 212–247.10.21136/CMJ.1985.102012Search in Google Scholar
[194] Schwarz, Š.: A combinatorial problem arising in finite Markov chains, Math. Slovaca 36 (1986), 199–210.Search in Google Scholar
[195] Schwarz, Š.: Powers of subsets in a finite semigroup, Semigroup Forum 51(1) (1995), 1–22.10.1007/BF02573616Search in Google Scholar
[196] Schwarz, Š.: Universal formulae of Euler-Fermat type for subsets of ℤ m, Collect. Math. 46(1–2) (1995), 183–193.Search in Google Scholar
[197] Schwarz, Š.—Krajňáková, D.: On totally non-commutative semigroups, Mat.-Fyz. Časopis Slovensk. Akad. Vied. 9(2) (1959), 92–100 (in Slovak).Search in Google Scholar
[198] Scorza, G.: Corpi Numerici ed Algebre, Principato, Messina, 1921.Search in Google Scholar
[199] Seidenberg, A.: Elements of the Theory of Algebraic Curves, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968.Search in Google Scholar
[200] Seneta, E.: Non-negative Matrices and Markov Chains. Springer Series in Statistics, Revised reprint of the second (1981) edition, Springer-Verlag, New York, 2006.10.1007/0-387-32792-4Search in Google Scholar
[201] Shajn, B. M.: On certain classes of semigroups of binary relations, Sib. Mat. J. 6 (1965), 616–635 (in Russian).Search in Google Scholar
[202] Shevrin, L. N.: Epigroups. In: Structural Theory of Automata, Semigroups, and Universal Algebra. Proceedings of the NATO Advanced Study Institute, Montreal, Quebec, Canada, July 7–18, 2003, Kluwer Academic Publishers, Dordrecht, 2005, pp. 331–380.10.1007/1-4020-3817-8_12Search in Google Scholar
[203] Snider, R. L.: Complemented hereditary radicals, Bull. Aust. Math. Soc. 4 (1971), 307–320.10.1017/S0004972700046669Search in Google Scholar
[204] Study, E.: Theorie der gemeinen und höheren complexen Grössen, Encykl. d. math. Wiss. 1 (1898), 147–183.10.1007/978-3-663-16017-5_4Search in Google Scholar
[205] Szalay, M.: On the maximal order in Sn and S*_n, Acta Arith. 37 (1980), 321–331.10.4064/aa-37-1-321-331Search in Google Scholar
[206] Szele, T.: Une généralisation de la congruence de Fermat, Mat. Tidsskr. B (1948), 57–59.Search in Google Scholar
[207] Šaĭn, B. M. Lectures on transformation semigroups, Special course, Izdat. Saratov. Univ., Saratov, 1970, 50 pp. (in Russian).Search in Google Scholar
[208] Šulka, R.: Radicals and topology in semigroups, Mat.-Fyz. Časopis Slovensk. Akad. Vied 15 (1965), 3–14.Search in Google Scholar
[209] Taber, H.: On the theory of matrices, Amer. J. Math. XII (1890), 337–396.10.2307/2369849Search in Google Scholar
[210] Vandiver, H. S.: The elements of a theory of abstract discrete semigroups, Vierteljschr. Naturforsch. Ges. Zürich 85 (1940), Beiblatt (Festschrift Rudolf Fueter), 71–86.Search in Google Scholar
[211] Walker, G. L.: Fermat’s theorem for algebras, Pacific J. Math. 4 (1954), 317–320.10.2140/pjm.1954.4.317Search in Google Scholar
[212] Wallace, A. D.: A note on mobs, II, Anais Acad. Brasil. Ci. 25 (1953), 335–336.10.1136/bmj.1.4805.336-aSearch in Google Scholar
[213] Weber, H.: Lehrbuch der Algebra, Vol. 2, 2nd ed., Friedr. Vieweg & Sohn, Braunschweig, 1899.Search in Google Scholar
[214] Wedderburn, J. H. M.: Lectures on Matrices. Colloq. Publ. 17, Amer. Math. Soc., New York, 1934.10.1090/coll/017Search in Google Scholar
[215] Weierstrass, K.: Zur Theorie der aus n Haupteinheiten gebildeten complexen Grössen, Nachr. Könogl. Gess. Wiss. Göttingen 10 (1884), 395–419.10.1017/CBO9781139567817.016Search in Google Scholar
[216] Wielandt, H.: Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642–648.10.1007/BF02230720Search in Google Scholar
[217] Zarelua, A. V.: On matrix analogs of Fermat’s little theorem, Math. Notes 79 (2006), 783–796.10.1007/s11006-006-0090-ySearch in Google Scholar
[218] Zariski, O.—Samuel, P.: Commutative Algebra, Vol. 1, The University Series in Higher Mathematics, D. van Nostrand Company, Inc., Princeton-Toronto-New York-London, 1958.Search in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Idempotents, group membership and their applications
- Residuation in non-associative MV-algebras
- An extension of F. Šik’s theorem on modular lattices
- Weak pseudo-BCK algebras
- Witt functor of a quadratic order
- A modification of a problem of Diophantus
- Cauchy problems involving a Hadamard-type fractional derivative
- Caratheodory’s solution of the Cauchy problem and a question of Z. Grande
- Sturm-Picone comparison theorems for nonlinear impulsive differential equations
- On oscillatory fourth order nonlinear neutral differential equations – III
- Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales
- On the representation of involutive Jamesian functions
- Refinements of the heinz inequalities for operators and matrices
- Generalizations of Reid inequality
- Probabilistic convergence transformation groups
- Cluster sets and topology
- Some characterizations for Markov processes as mixed renewal processes
- Equivalent conditions of complete convergence for weighted sums of ANA random variables
Articles in the same Issue
- Idempotents, group membership and their applications
- Residuation in non-associative MV-algebras
- An extension of F. Šik’s theorem on modular lattices
- Weak pseudo-BCK algebras
- Witt functor of a quadratic order
- A modification of a problem of Diophantus
- Cauchy problems involving a Hadamard-type fractional derivative
- Caratheodory’s solution of the Cauchy problem and a question of Z. Grande
- Sturm-Picone comparison theorems for nonlinear impulsive differential equations
- On oscillatory fourth order nonlinear neutral differential equations – III
- Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales
- On the representation of involutive Jamesian functions
- Refinements of the heinz inequalities for operators and matrices
- Generalizations of Reid inequality
- Probabilistic convergence transformation groups
- Cluster sets and topology
- Some characterizations for Markov processes as mixed renewal processes
- Equivalent conditions of complete convergence for weighted sums of ANA random variables