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• 3.?????????,??????????????????
• ??????
• 1 J. P. M. Binet, Me moire sur un systµeme de
  formules analytiques, et leur application µa des
  considerations geometriques, J. Ec. Polyt.9
  (1812)280-302
• 2 Z. Q. Cao, K. H. Kim, F.W.Roush, Incline
  Algebra andApplications, John Wiley, New York,
  1984

29

• 3 A. L. Cauchy, Memoire sur les fonctions qui
  ne peuvent obtenir que deux valeurs egales et de
  signes contraires par suite des traspositions
  operees entre les variables qu'elles renferment,
  J. Ec. Polyt. 10(1812) 29-11220
• 4 R. A. Cuninghame-Green, Minimax algebra,
  Lecture Notes in Economics and Mathematical
  Systems 166, Springer-Verlag, Berlin, 1979
• 5 J. S. Duan, The transitive closure,
  convergence of powers and adjoint of generalized
  fuzzy matrices. Fuzzy Sets and Systems
  145(2004)301-311

30

• 6 J. S. Golan, Semirings and Their
  Applications, Kluwer Academic Publishers,1999
• 7 S. C. Han, H. X. Li, Invertible incline
  matrices and Cramer's rule over inclines, Linear
  Algebra and its Applications 389(2004)121-138
• 8 Y. Huang, Y. J. Tan, A problem on incline
  matrices, J. of Fuzhou University 37(2009)12-18
  (in Chinese)
• 9 J. B. Kim, A. Baartmans, N. S. Sahadin,
  Determinant theory for fuzzy matrices, Fuzzy Sets
  and Systems 29(1989)349-356.

31

• 10 B. R. Mcdonald, Linear Algebra over
  Commutative Rings, Marcel Dekker, INC. New
  York,1984.
• 11 H. Minc, Permanents, Addison-Wesley
  Publishing Company, Massachusetts, U. S. A. 1978.
• 12 P. L. Poplin, R. E. Hartwig, Determinantal
  identities over commutative semirings, Linear
  Algebra and its Applications 387(2004)99-132
• 13 M. Z. Ragab, E. G. Emam, The determinant and
  adjoint of a square fuzzy matrix, Fuzzy Sets and
  Systems 61(1994)297-307

32

• 14 Y. J. Tan, On invertible matrices over
  antirings, Linear Algebra and its Applications
  423(2007)428-444
• 15 Y. J. Tan, On invertible matrices over
  commutative semirings, Linear and Multilinear
  Algebra 61(2013)710-714
• 16 Z. J. Tian, K. M. Yan, D. G. Li, H. Zhao,
  Determinant of matrices over completely
  distributive lattices, J. Gansu Univ. Technol. 28
  (4) (2002) 115-118 (in Chinese)

33

• 17 E. M. Vechtomov, Two general structure
  theorems on submodules, Abelian Groups and
  Modules (in Russion), Tomsk State University,
  Tomsk, No15(2000)17-23
• 18 K. L. Zhang, Determinant theory for
  D01-lattices, Fuzzy Sets and Systems
  62(1994)347-353
• 19 U. Zimmermann, Linear and combinatorial
  optimization in ordered algebraic structures,
  Annals of Discrete Mathematics, Vol. 10, North
  Holland, 1981.2

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Thankyou!