Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
ring with identity; homomorphism; one-sided ideal; two-sided ideal; module; bimodule
ring with identity; homomorphism; one-sided ideal; two-sided ideal; module; bimodule
Summary:
The main result of this paper is the introduction of a notion of a generalized $R$-Latin square, which includes as a special case the standard Latin square, as well as the magic square, and also the double stochastic matrix. Further, the algebra of all generalized Latin squares over a commutative ring with identity is investigated. Moreover, some remarkable examples are added.
References:
[1] Andrew, W. S.: Magic Squares and Cubes. Dover, New York (1960). MR 0114763
[2] Birkhoff, G.: Tres observaciones sobre el algebra lineal. Rev., Ser. A, Univ. Nac. Tucuman 5 (1946), 147-150. MR 0020547
[3] Cayley, A.: On the theory of groups. Proc. London Math. Soc. 9 (1877/78), 126-133.
[4] Davis, P.: Circulant Matrices. London (1970).
[5] Dènes, J., Keedwell, A. D.: Latin Squares and Their Applications. Akadémiai Kiadó, Budapest, (1974). MR 0351850
[6] Euler, L.: Recherches sur une nouvelle espace de carrés magiques. Verh. Zeeuwsch. Genootsch. Wetensch. Vlissengen 9 (1782), 85-239.
[7] Fano, G.: Sui postulati fundamentali della geometria proiectiva. Giorn. Math. 30 (1892), 106-112.
[8] Fisher, R. A.: The Design of Experiments. Olivier et Boyd, Edinburgh (1937).
[9] Hall, jun., M.: Combinatorial Theory. Blaisdell Publ. Comp., Toronto (1967). Zbl 0196.02401
[10] Herstein, I. N.: Rings with Involutions. Univ. of Chicago Press (1976). MR 0442017
[11] Hungerford, T. V.: Algebra. Springer, New York (1980). MR 0600654 | Zbl 0442.00002
[12] Kárteszi, F.: Introduction to Finite Geometries. Akademiai Kiadò, Budapest (1976). MR 0423175
[13] Kasch, F.: Moduln und Ringe. Teubner, Stuttgart (1977). MR 0429963 | Zbl 0343.16001
[14] Katrnoška, F.: Logics that are generated by idempotents. Lobachevskij J. Math. 15 (2004), 11-19. MR 2120697 | Zbl 1060.15018
[15] Katrnoška, F.: Latin squares and the genetic code. Pokroky Mat. Fyz. Astronom. 52 (2007), 177-187 Czech. Zbl 1265.05078
[16] Kostrikin, A. I., Shafarewich, I. R.: Algebra I. Springer, Berlin (1990).
[17] Marcus, M.: Some properties and applications of doubly stochastic matrices. Amer. Math. Monthly 67 (1960), 215-221. DOI 10.2307/2309679 | MR 0118732 | Zbl 0092.01601
[18] Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston (1964). MR 0162808 | Zbl 0126.02404
[19] Moufang, R.: Zur Struktur von alternativ Körpern. Math. Ann. 110 (1935), 416-430. DOI 10.1007/BF01448037 | MR 1512948
[21] Schafer, R. D.: Structure of genetic algebras. J. Amer. Math. 71 (1949), 121-135. DOI 10.2307/2372100 | MR 0027751 | Zbl 0034.02004
[22] Singer, J.: A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. 43 (1938), 377-385. DOI 10.1090/S0002-9947-1938-1501951-4 | MR 1501951 | Zbl 0019.00502
[23] Singer, J.: A class of groups associated with Latin squares. Amer. Math. Monthly 67 (1960), 235-240. DOI 10.2307/2309683 | MR 0124227 | Zbl 0096.01202
[24] Steinfeld, O.: Über die Struktursätze der Semiringe. Acta Math. Acad. Scient. Hung. 10 (1959), 149-155. DOI 10.1007/BF02063296 | MR 0108523 | Zbl 0087.02801
[25] Wiegandt, R.: Über die Struktursätze der Halbringe. Ann. Univ. Sci. Budap. Rolando Eötvös, Sec. Math. 5 (1962), 51-68. MR 0148712 | Zbl 0123.00901
[26] Wörz-Busekros, A.: Algebras in Genetics. Springer, Berlin, Heidelberg, New York (1980). MR 0599179
Search
Partner of
