JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2015, Vol. 50 ›› Issue (06): 45-52.doi: 10.6040/j.issn.1671-9352.0.2014.229

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Dimensions of semilinear spaces over commutative semirings

ZHANG Hou-jun, CHU Mao-quan   

  1. School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, Anhui, China
  • Received:2014-05-20 Revised:2014-11-05 Online:2015-06-20 Published:2015-07-31

Abstract: The dimensions of semilinear spaces over commutative semirings L are investigated. Some necessary and sufficient conditions that dim(Vn)=n are given, and the relationship between Vn and V1 are obtained, where Vn and V1 are finite dimensional semilinear spaces over L. Moreover, the concepts of semilinear transformation A, and the range A(Vn) and nuclear A-1(0) of A are introduced and the equation dim(A(Vn))+dim(A-1(0))=dim(Vn) is proved.

Key words: commutative semiring, semilinear spaces, dimensions, semilinear transformation

CLC Number: 

  • O153.3
[1] CUNINGHAME-GREEN R A. Minimax algebra[M]// Lecture Notes in Economics and Mathematical Systems. Berlin: Springer, 1979.
[2] BUTKOVIC P. Max-algebra: the linear algebra of combinatorics[J]. Linear Algebra Appl, 2003, 367:313-335.
[3] CAO Z Q, KIM K H, ROUSH F W. Incline algebra and applications[M]. New York: John Wiley, 1984.
[4] CECHLAROVA K, PLAVKA J. Linear independence in bottleneck algebras[J]. Fuzzy Sets and Systems, 1996, 77:337-348.
[5] KIM K H, ROUSH F W. Generalized fuzzy matrices[J]. Fuzzy Sets and Systems, 1980, 4:293-315.
[6] DI-NOLAR A, LETTIERI A, PERFILIEVA I. Algebraic analysis of fuzzy systems[J]. Fuzzy Sets and Systems, 2007, 158:1-22.
[7] ZHAO Shan, WANG Xueping. Invertible matrices and semilinear spaces over commutative semirings[J]. Information Sciences, 2010, 180:5115-5124.
[8] ZHAO Shan, WANG Xueping. Bases in semilinear spaces over join-semirings[J]. Fuzzy Sets and Systems, 2011, 182:93-100.
[9] SHU Qianyu, WANG Xueping. Bases in semilinear spaces over zerosumfree semirings[J]. Linear Algebra Appl, 2011, 435:2681-2692.
[10] SHU Qianyu, WANG Xueping. Standard orthogonal vectors in semilinear spaces and their applications[J]. Linear Algebra Appl, 2012, 437:2733-2754.
[11] SHU Qianyu, WANG Xueping. Dimensions of L-semilinear spaces over zerosumfree semirings[C]// IFSA World Congress and NAFIPS Annual Meeting. Edmonton, Canada: IEEE, 2013: 35-40.
[12] GOLAR J S. Semirings and their applications[M]. Dordrecht: Kluwer Academic Publishers, 1999.
[13] REUTENAUER C, STRAUBING H. Inversion of matrices over a commutative semiring[J]. J Algebra, 1984, 88:350-360.
[14] TAN Yijia. Bases in semimodules over commutative semirings[J]. Linear Algebra Appl, 2014, 443:139-152.
[15] KANAN A M, PETROVIC Z Z. Note on cardinality of bases in semilinear spaces over zerosumfree semirings[J]. Linear Algebra Appl, 2013, 439:2795-2799.
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