The normalization of rectangular algebras

• Autorzy: Vanžurová A.
• Języki publikacji: EN

Abstrakty

EN

Our aim is to investigate the normalization (nilpotent shift) (RAr}* of the variety RAr of rectangular algebras of a finite type r. It can be regarded as a common generalization of various algebras studied by different authors, namely of rectangular algebras, [P&R 92, 93], rectangular bands, [Gou], medial grupoids, [J&K], or generalized diagonal algebras, [P 64, 66a, b]. We will determine all subdirectly irreducible algebras in the normalization of the variety of rectangular algebras, give a normal form for terms in the normalization and an algorithm for finding a subdirect decomposition of algberas belonging to the normalization.

Słowa kluczowe

EN

projection; rectangular algebra; diagonal algebra; normalization

PL

algebra diagonalna

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2003
• Tom: Vol. 36, nr 1
• Strony: 11--28
• Opis fizyczny: Bibliogr. 25 poz., tab.

Twórcy

autor

Vanžurová A.

• Department of Algebra and Geometry, Palacký University, Tomkova 40, 779 00 Olomouc, Czech Republic

Bibliografia

• [Ch 95a] I. Chajda, Congruence properties of algebras in nilpotent shifts of varieties, General Alg. and Discrete Math., K. Denecke, O. Lüders (eds.), Heldermann-Verlag, Berlin (1995), 35-46.
• [Ch 95b] I. Chajda, Normally presented varieties, Alg. Universalis 34 (1995), 327-335.
• [Ch&G 99] I. Chajda and E. Graczyńska, Algebras presented by normal identities, Acta Univ. Palack. Olomuc., Fac. Rer. Nat. 38 (1999), 45-58.
• [ChHPR] I. Chajda, R. Halaš, A. G. Pinus and I. G. Rosenberg, Duality of normally presented varieties, Internat. Jour. of Alg. and Comput., 10, No. 5 (2000), World Scient. Publ. Comp., 651-664.
• [D&P 88] K. Denecke and R. Pöschel, The Characterisation of primal algebras by hyperidentities, Contribuitons to General Algebra 6, Verlag Hölder-PichlerTempsky, Wien, 1988, pp. 67-87.
• [DLPS 91] K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, hyperequational classes and clone congruences, Contribuitons to General Algebra 7, Verlag Hölder-Pichler-Tempsky, Wien, 1991, 97-118.
• [Gou] M. Gould, Rectangular bands in universal algebras, Acta Sei. Math. 44 (1982), 271-279.
• [Gr 88] E. Graczyńska, On connection between identities and hyperidentities, Bull. Sect. Logic 17 (1988), 34-41.
• [Gr 98] E. Graczyńska, Operators on Varieties, Gen. Alg. and Appl. (1998), Politechnika Opolska, Opole.
• [Ih] Th. Ihringer, Allgemeine Algebra, Teubner, Stuttgart, 1988.
• [J&K] J. Ježek and T. Kepka, Medial Grupoids, Academia, Praha, 1983.
• [Ke] K. A. Kearnes, Semilattice models I: the associated semiring, Algebra Universalis 34 (1995), 220-272.
• [K&V] J. Klouda and A. Vanžurová, On a general construction of diagonal algebras, Demonstratio Math. 33 (2000), 223-230.
• [L] E. S. Liapin, Semigroups, 1960 (in Russian).
• [M 73] I. I. Mel'nik, Nilpotent sfifts of varieties, Mat. Zametki 14 (1973), 703-712 (in Russian).
• [P 64] J. Płonka, Diagonal algebras and algebraic independence, Bull. Acad. Polon. Sci. (1964), 729-733.
• [P 66a] J. Płonka, Diagonal algebras, Fund. Math. 58 (1966), 309-321.
• [P 66b] J. Płonka, Remarks on diagonal and generalized diagonal algebras, Colloq. Math. 15 (1966), 19-23.
• [P&R 92] R. Pöschel and M. Reichel, Rectangular algebras, Proc. 22nd Internat. Sympos. on Multiple-valued Logic, Sendai (Japan), IEEE Comp. Soc. Press Los Alamitos (1992), California, 255-260.
• [P&R 93] R. Pöschel and M. Reichel, Projection algebras and rectangular algebras, K. Denecke, H.-J. Vogel (eds.), General Alg. and Appl., Research and Exposition in Math. Vol. 20, Heldermann Verlag, Berlin, 1993, 180-194.
• [R 92] A. Romanowska, An introduction to the theory of modes and modals, Contemp. Math. 131 (1992), 241-262.
• [R&S] A. B. Romanowska and J. D. H. Smith, Modal Theory, An Algebraic Approach to Order, Geometry, and Convexity, Heldermann Verlag, Berlin 1985.
• [SI 92] J. Šlapal, A note on diagonality of algebras, Math. Nachr. 158 (1992), 195-197.
• [SI 94] J. Šlapal, On exponentiation of n-ary algebras, Acta Math. Hungar. 63 (4) (1994), 313-322.
• [Sm 99] J. D. H. Smith, Modes and modals, Discuss. Math., Algebra and Stochastic Methods 19 (1999), 9-40.