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STRUCTURES OF IDEMPOTENT MATRICES OVER CHAIN SEMIRINGS

  • Published : 2007.11.30

Abstract

In this paper, we have characterizations of idempotent matrices over general Boolean algebras and chain semirings. As a consequence, we obtain that a fuzzy matrix $A=[a_{i,j}]$ is idempotent if and only if all $a_{i,j}$-patterns of A are idempotent matrices over the binary Boolean algebra $\mathbb{B}_1={0,1}$. Furthermore, it turns out that a binary Boolean matrix is idempotent if and only if it can be represented as a sum of line parts and rectangle parts of the matrix.

Keywords

semiring;idempotent;frame;rectangle part;line part

References

  1. L. B. Beasley and N. J. Pullman, Linear operators strongly preserving idempotent ma- trices over semirings, Linear Algebra Appl. 160 (1992), 217-229 https://doi.org/10.1016/0024-3795(92)90448-J
  2. J. S. Golan, Semirings and their applications, updated and expanded version of the theory of semirings, with applications to mathematics and theoretical computer science, Kluwer Academic Publishers, Dordrecht, 1999
  3. D. A. Gregory and N. J. Pullman, Semiring rank : Boolean rank and nonnegative rank factorizations, J. Combin. Inform. System Sci. 8 (1983), no. 3, 223-233
  4. S. Z. Song and K. T. Kang, Types and enumeration of idempotent matrices, Far East J. Math. Sci. 3 (2001), no. 6, 1029-1042
  5. S. Z. Song, K. T. Kang, and L. B. Beasley, Idempotent matrix preservers over Boolean algebras, J. Korean Math. Soc. 44 (2007), no. 1, 169-178 https://doi.org/10.4134/JKMS.2007.44.1.169
  6. R. B. Bapat, S. K. Jain, and L. E. Snyder, Nonnegative idempotent matrices and minus partial order, Linear Algebra Appl. 261 (1997), 143-154 https://doi.org/10.1016/S0024-3795(96)00364-3
  7. S. Kirkland and N. J. Pullman, Linear operators preserving invariants of non-binary matrices, Linear and Multilinear Algebra 33 (1993), no. 3-4, 295-300
  8. V. N. Kolokoltsov and V. P. Maslov, Idempotent Analysis and its Applications, Mathe-matics and its Applications, 401, Dordrecht: Kluwer Academic Publishers, 1997

Cited by

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  2. On Decompositions of Matrices over Distributive Lattices vol.2014, 2014, https://doi.org/10.1155/2014/202075
  3. Idempotent matrices over antirings vol.431, pp.5-7, 2009, https://doi.org/10.1016/j.laa.2009.03.035