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Some Conditions for Matrices over an Incline To Be Invertible and General Linear Group on an Incline

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Abstract

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GL n (L) = PL n (L) for any n (n ≥ 2), in which GL n (L) is the group of all n × n invertible matrices over L and PL n (L) is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.

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References

  1. Cao, Z. Q., Kim, K. H., Roush, F. W.: Incline Algebra and Applications, John Wiley, New York, 1984

  2. Kim, K. H., Roush, F. W.: Inclines and incline matrices: a survey. Linear Algebra Appl., 379, 457–473 (2004)

    Article  MathSciNet  Google Scholar 

  3. Give'on, Y.: Lattice matrices. Information and Control, 7, 477–484 (1964)

    Article  MathSciNet  Google Scholar 

  4. Skornyakov, L. A.: Invertible matrices over distributive structures (in Russian). Sibirsk. Mat. Zh., 27(2), 182–185 (1986)

    MathSciNet  Google Scholar 

  5. Zhao, C. K.: Invertible conditions for a matrix over distributive lattice (in Chinese). Acta Sci. Natur. Univ. NeiMonggol, 22(4), 477–480 (1991)

    Google Scholar 

  6. Duan, J. S.: The ranks of L–fuzzy regular matrices and invertible matrices (in Chinese). J. Shandong Univ., 36(3), 251–255 (2001)

    Google Scholar 

  7. Tan, Y. J.: The semigroup of Hall matrices over distributive lattices. Semigroup Forum., 61(2), 303–314 (2000)

    Article  MathSciNet  Google Scholar 

  8. Cheng, Y. J.: Residue semislops (in Chinese). Natur. Sci. J. Xiangtan Univ., 15(3), 8–13 (1993)

    Google Scholar 

  9. Luo, M. K., Li, S. T.: Connectedness and anti–diamond–type lattices (in Chinese). J. Sichuan Univ., 36(5), 807–810 (1999)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Mechanics, Kim Il Sung University, Pyongyang, D. P. R. Korea

    Song Chol Han

  2. Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China

    Song Chol Han

  3. Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China

    Hong Xing Li

Authors
  1. Song Chol Han
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  2. Hong Xing Li
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Correspondence to Song Chol Han or Hong Xing Li.

Additional information

Supported by National Natural Science Foundation of China (60174013), Research Foundation for Doctoral Program of Higher Education (20020027013), Science and Technology Key Project Foundation of Ministry of Education (03184) and Major State Basic Research Development Program of China (2002CB312200)

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Han, S.C., Li, H.X. Some Conditions for Matrices over an Incline To Be Invertible and General Linear Group on an Incline. Acta Math Sinica 21, 1093–1098 (2005). https://doi.org/10.1007/s10114-004-0497-x

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  • DOI: https://doi.org/10.1007/s10114-004-0497-x

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