偏序半环的偏序扩张
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摘要
偏序代数理论，是重要的代数学分支，许多专家学者对其进行了深入细致的研究。本文主要研究了偏序半环的偏序扩张和有限全序扩张，并得到了一些新的结果。
本文的主要内容分为三章。第一章介绍了本文的选题背景及课题意义，介绍了偏序半环、拟序、拟链、半格与格及其同态与序同态等概念与有关基础知识。第二章以偏序半环的偏序扩张为中心，引入了偏序半环的半拟序与半拟链的概念并得到了它们的一些基本性质，给出了偏序半环的偏序扩张的构造方法，得到了偏序半环能够偏序扩张的充分条件。第三章以偏序半环的有限全序扩张为中心，给出了偏序半环的有限全序扩张的概念和偏序半环能进行有限全序扩张的充分必要条件。
Ordered algebric theory is a very important branch of algebra. Many experts and scholars have investigated it thoroughly and painstakingly. In this paper, we mainly study extensions of partial order and finitely totally order for partially ordered semirings, and obtain some new results.
This dissertation can be divided into three parts. The first chapter, we introduce the history of semigroups, semirings, partially ordered semigroups, partially ordered semirings. we emphasize that the thesis is mainly concerned with several important papers arising in extensions of partial order for ordered algebraic theory. Also, we introduce the basic notions and results of partially ordered semirings, pseudoorder, pseudochain, semi-lattices, lattices, isotonic homomorphism and homomorphism of them. The second chapter, we introduce the notions of semi-pseudoorderσand semi-pseudochain modσon the partially ordered semirings and give some basic properties of them. we also give a constructive method of extensions of partial order for partially ordered semirings. Besides, we get sufficient condition of extensions of partial order for partially ordered semirings by semi-pseudochain modσ, and obtain some interesting results. In chapter 3. we give the notion of extensions of finitely totally order for partially ordered semirings, Besides, we discuss sufficient and necessary conditions on extensions of finitely totally order for partially ordered semirings.
本文的主要内容分为三章。第一章介绍了本文的选题背景及课题意义，介绍了偏序半环、拟序、拟链、半格与格及其同态与序同态等概念与有关基础知识。第二章以偏序半环的偏序扩张为中心，引入了偏序半环的半拟序与半拟链的概念并得到了它们的一些基本性质，给出了偏序半环的偏序扩张的构造方法，得到了偏序半环能够偏序扩张的充分条件。第三章以偏序半环的有限全序扩张为中心，给出了偏序半环的有限全序扩张的概念和偏序半环能进行有限全序扩张的充分必要条件。
Ordered algebric theory is a very important branch of algebra. Many experts and scholars have investigated it thoroughly and painstakingly. In this paper, we mainly study extensions of partial order and finitely totally order for partially ordered semirings, and obtain some new results.
This dissertation can be divided into three parts. The first chapter, we introduce the history of semigroups, semirings, partially ordered semigroups, partially ordered semirings. we emphasize that the thesis is mainly concerned with several important papers arising in extensions of partial order for ordered algebraic theory. Also, we introduce the basic notions and results of partially ordered semirings, pseudoorder, pseudochain, semi-lattices, lattices, isotonic homomorphism and homomorphism of them. The second chapter, we introduce the notions of semi-pseudoorderσand semi-pseudochain modσon the partially ordered semirings and give some basic properties of them. we also give a constructive method of extensions of partial order for partially ordered semirings. Besides, we get sufficient condition of extensions of partial order for partially ordered semirings by semi-pseudochain modσ, and obtain some interesting results. In chapter 3. we give the notion of extensions of finitely totally order for partially ordered semirings, Besides, we discuss sufficient and necessary conditions on extensions of finitely totally order for partially ordered semirings.
引文
[1] Stephen C.Kleene, Representation of events in nerve nets and finite automata, in C. E. Shannon, J. McCarthy (eds.): Automata Studies, Princeton Univ. Press, Princeton, 1956.
[2] Richard Dedekind, Uber die Theorie der ganzen algebraiscen Zahlen, Supplement Ⅺ to P.G.Lejeune Dirichlet: Vorlesung Uber Zahlentheorie 4 Aufl. Druck and Verlag, Braunschweig, 1894.
[3] Arto Salomaa, Matti Soittola, Automata-theoretic Aspects of Formal Power Series. Springer-Verlag, Berlin, 1978.
[4] J.W. de Bakker, D.Scott, A theory of Programs. IBM, Vienna, 1969.
[5] Cyril Allauzen, Mehryar Mohri, Efficient algorithms for testing the twin primes property. J. Autom. Lang. Combin. 8 (2003), 117-144.
[6] Raymond A, Cuninghame-Green, Minimax Algebra. Lecture Notes in Economics and Mathematical Systems vol. 166, Springer-Verlag, Berlin, 1979.
[7] Vassili N, Kolokol'tsov, Victor Maslov, Idempotent Analysis and Applications. Kluwer, Dordrecht, 1997.
[8] Michel Gondran. Michel Minoux, Graphs and Algorithms. Wiley-Interscience. New York, 1984.
[9] Wolfgang Wechler. R-fuzzy computation. J. Math. Anal. Appl. 115 (1986). 225-232.
[10] Jean-Louis Boimond, L. Haroudin, P.Chiron, A modelling method of SISO discrete-event systems in max-algebra. Proceedings of ECC 95, Rome. 1995.
[11] Gary Miller. V. Ramacandran. E. Kaltofen. Efficient parallel evaluation of straight-line code and arithmetic circuit. SIAM J. Comput. 17 (1988) 687-695.
[12] Gary Miller, Shang-Hua Teng, The dynamic parallel complexity of computational circuits. SIAM J. Comput. 28 (1999) 1664-1688.
[13] Jonathan S. Golan, The Theory of Semirings with Applications and Theoretical Computer Science. Longman Science Technology, New York, (1992).
[14] Jonathan S. Golan. Semirings and Their Applications. Kluwer, Dordrecht. 1999.
[15] Jonathan S. Golan, Semirings and Affine Equations over Them: Theory and Applications. Kluwer, Dordrecht, 2003.
[16] Jonathan S. Golan, Power Algebras over Semirings. Kluwer, Dordrecht, 1999.
[17] T.S.Blyth and M.F.Janowtz, Residuation Theory. Pergamon, 1972.
[18] N.Kehayopulu and M.Tsingelis, The embedding of an ordered semigroup in a simple one with identity. Semigroup Forum. 53 (1996). 346-350.
[19] N.Kehayopulu and M.Tsingelis, N-subsets of ordered semigroups. In: E. S. Lyapin(ed.), Decompositions and homomorphic mappings of semigroups, Interuniversity collect of scientific words. Sankt-Peterburge: Obrazovanie. 1992.56-63.
[20] N.Kehayopulu and M.Tsingelis, On separative ordered semigroups. Semigroup Forum, 56 (1998), 187-196.
[21] N.Kehayopulu and M.Tsingelis, On the decomposition of prime ideals of ordered semigroups into their N-classes. Semigroup Forum, 47 (1993), 393-395.
[22] N.Kehayopulu, S.Lajos and M.Tsingelis, Note on prime, weakly prime ideals in ordered semigroups. Pure Math, and Appl, 5 (1994), No. 2, 293-297.
[23] N.Kehayopulu and M.Tsingelis, Pseudo-order in ordered semigroups. Semigroup Forum, 50 (1995), 389-392.
[24] N.Kehayopulu and M.Tsingelis, On weakly commutative ordered semigroups. Semigroups Forum, 56 (1998), 32-35.
[25] N.Kehayopulu and M.Tsingelis, Acharacterization of strongly regular ordered semigroups. Math. Japonica 48 (1998), 213-215.
[26] N.Kehayopulu and M.Tsingelis, On subdirectly irreducible ordered semigroups. Semigroup Forum, 50 (1995), 161-177.
[27] X.Y.Xie, Introduction To Ordered Semigroups. Beijing: KeXue Press, 2001. (in Chinese)
[28] Georg Karner, On limits in complete semirings. Semigroup Forum, 45 (1992). 148-165.
[29] Martin Goldstern, Vervollstandigung von Halbringen. Diplomarbeit, Technische Universitat Wien, 1985.
[30] K.A.Kearnes, Semilattice modes I: the associated semiring. Algebra Universalis, 34 (1995), 220-272. Fuchs
[31] FUCHS L, Partially Ordered Algebraic Systems. Pergaman Aress, 1963.
[32] Nakada O, Partially Ordered Ablian Semigroups I . J. Fac. Sct. Hokkaido Univ. 11 (1951), 181-189.
[33] Nakada O, Partially Ordered Ablian Semigroups II . J. Fac. Sct. Hokkaido Univ. 12 (1952), 73-86.
[34] Dubreil-Jacotin, Sures O-Bands. Semigroup Forum, 3 (1971), 156-159.
[35] A.J.Hulin, Extensions of Ordered Semigroups. Semigroups Forum. 2 (1971), 336-342.
[36] A.J.Hulin, Extensions of Ordered Semigroups . Czech. Math. J. 26 (1976). 1-12.
[37] X.Y.Xie, An Extensions Of Partial Orders Of Commutative Partially Ordered Semigroups. Journal of Mathematical Research and Exposition. Vol. 25 (4) (2005). 676-682.
[38] B.A.Davey. H.A.Priestley. Introduction to Lattices and Order. Cambridge University Press. 8 (2003). 117-144.
[39] HOWIE J M, An Introduction To Semigroups Theory. London: Acad. Press, 1976.
[2] Richard Dedekind, Uber die Theorie der ganzen algebraiscen Zahlen, Supplement Ⅺ to P.G.Lejeune Dirichlet: Vorlesung Uber Zahlentheorie 4 Aufl. Druck and Verlag, Braunschweig, 1894.
[3] Arto Salomaa, Matti Soittola, Automata-theoretic Aspects of Formal Power Series. Springer-Verlag, Berlin, 1978.
[4] J.W. de Bakker, D.Scott, A theory of Programs. IBM, Vienna, 1969.
[5] Cyril Allauzen, Mehryar Mohri, Efficient algorithms for testing the twin primes property. J. Autom. Lang. Combin. 8 (2003), 117-144.
[6] Raymond A, Cuninghame-Green, Minimax Algebra. Lecture Notes in Economics and Mathematical Systems vol. 166, Springer-Verlag, Berlin, 1979.
[7] Vassili N, Kolokol'tsov, Victor Maslov, Idempotent Analysis and Applications. Kluwer, Dordrecht, 1997.
[8] Michel Gondran. Michel Minoux, Graphs and Algorithms. Wiley-Interscience. New York, 1984.
[9] Wolfgang Wechler. R-fuzzy computation. J. Math. Anal. Appl. 115 (1986). 225-232.
[10] Jean-Louis Boimond, L. Haroudin, P.Chiron, A modelling method of SISO discrete-event systems in max-algebra. Proceedings of ECC 95, Rome. 1995.
[11] Gary Miller. V. Ramacandran. E. Kaltofen. Efficient parallel evaluation of straight-line code and arithmetic circuit. SIAM J. Comput. 17 (1988) 687-695.
[12] Gary Miller, Shang-Hua Teng, The dynamic parallel complexity of computational circuits. SIAM J. Comput. 28 (1999) 1664-1688.
[13] Jonathan S. Golan, The Theory of Semirings with Applications and Theoretical Computer Science. Longman Science Technology, New York, (1992).
[14] Jonathan S. Golan. Semirings and Their Applications. Kluwer, Dordrecht. 1999.
[15] Jonathan S. Golan, Semirings and Affine Equations over Them: Theory and Applications. Kluwer, Dordrecht, 2003.
[16] Jonathan S. Golan, Power Algebras over Semirings. Kluwer, Dordrecht, 1999.
[17] T.S.Blyth and M.F.Janowtz, Residuation Theory. Pergamon, 1972.
[18] N.Kehayopulu and M.Tsingelis, The embedding of an ordered semigroup in a simple one with identity. Semigroup Forum. 53 (1996). 346-350.
[19] N.Kehayopulu and M.Tsingelis, N-subsets of ordered semigroups. In: E. S. Lyapin(ed.), Decompositions and homomorphic mappings of semigroups, Interuniversity collect of scientific words. Sankt-Peterburge: Obrazovanie. 1992.56-63.
[20] N.Kehayopulu and M.Tsingelis, On separative ordered semigroups. Semigroup Forum, 56 (1998), 187-196.
[21] N.Kehayopulu and M.Tsingelis, On the decomposition of prime ideals of ordered semigroups into their N-classes. Semigroup Forum, 47 (1993), 393-395.
[22] N.Kehayopulu, S.Lajos and M.Tsingelis, Note on prime, weakly prime ideals in ordered semigroups. Pure Math, and Appl, 5 (1994), No. 2, 293-297.
[23] N.Kehayopulu and M.Tsingelis, Pseudo-order in ordered semigroups. Semigroup Forum, 50 (1995), 389-392.
[24] N.Kehayopulu and M.Tsingelis, On weakly commutative ordered semigroups. Semigroups Forum, 56 (1998), 32-35.
[25] N.Kehayopulu and M.Tsingelis, Acharacterization of strongly regular ordered semigroups. Math. Japonica 48 (1998), 213-215.
[26] N.Kehayopulu and M.Tsingelis, On subdirectly irreducible ordered semigroups. Semigroup Forum, 50 (1995), 161-177.
[27] X.Y.Xie, Introduction To Ordered Semigroups. Beijing: KeXue Press, 2001. (in Chinese)
[28] Georg Karner, On limits in complete semirings. Semigroup Forum, 45 (1992). 148-165.
[29] Martin Goldstern, Vervollstandigung von Halbringen. Diplomarbeit, Technische Universitat Wien, 1985.
[30] K.A.Kearnes, Semilattice modes I: the associated semiring. Algebra Universalis, 34 (1995), 220-272. Fuchs
[31] FUCHS L, Partially Ordered Algebraic Systems. Pergaman Aress, 1963.
[32] Nakada O, Partially Ordered Ablian Semigroups I . J. Fac. Sct. Hokkaido Univ. 11 (1951), 181-189.
[33] Nakada O, Partially Ordered Ablian Semigroups II . J. Fac. Sct. Hokkaido Univ. 12 (1952), 73-86.
[34] Dubreil-Jacotin, Sures O-Bands. Semigroup Forum, 3 (1971), 156-159.
[35] A.J.Hulin, Extensions of Ordered Semigroups. Semigroups Forum. 2 (1971), 336-342.
[36] A.J.Hulin, Extensions of Ordered Semigroups . Czech. Math. J. 26 (1976). 1-12.
[37] X.Y.Xie, An Extensions Of Partial Orders Of Commutative Partially Ordered Semigroups. Journal of Mathematical Research and Exposition. Vol. 25 (4) (2005). 676-682.
[38] B.A.Davey. H.A.Priestley. Introduction to Lattices and Order. Cambridge University Press. 8 (2003). 117-144.
[39] HOWIE J M, An Introduction To Semigroups Theory. London: Acad. Press, 1976.