Acta mathematica scientia, Series B >
THE LARGEST SOLUTION OF LINEAR EQUATION OVER THE COMPLETE HEYTING ALGEBRA
Received date: 2007-09-02
Revised date: 2008-04-18
Online published: 2010-05-20
Supported by
This work was supported by the NNSF (10471035, 10771056) of China
Let (L, ≤, ∨, ∧) be a complete Heyting algebra. In this article, the linear system Ax=b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by ∨ and ∧ respectively, is studied. We obtain: (i) the necessary and sufficient conditions for S(A, b)≠Ø; (ii) the necessary conditions for |S(A, b)| =1. We also obtain the vector x ∈ Ln and prove that it is the largest
element of S(A, b) if S(A, b)≠Ø.
Key words： Complete lattice; Heyting algebra; linear syste; minimal covering
ZHOU Jing-Lei , LI Qiang-Guo . THE LARGEST SOLUTION OF LINEAR EQUATION OVER THE COMPLETE HEYTING ALGEBRA[J]. Acta mathematica scientia, Series B, 2010 , 30(3) : 810 -818 . DOI: 10.1016/S0252-9602(10)60080-X
[1] Gierz G, Hofmann K H, Keimel K, Lawson J D, Mislove M, Scott D S. Continuous lattice and Domains. London: Cambridge Univ Press, 2003
[2] Davey B A, Priestley H A. Introduction to Lattice and Order. London: Cambridge Univ Press, 2002
[3] Katarina Cechlarova. Unique solvability of max-min fuzzy equations and strong regularity of matrices over fuzzy algebra. Fuzzy Sets and Systems, 1995, 75: 165--177
[4] Cuninghame-Green R A. Minimax Algebra. Lecture Notes in Econ and Math Systems, Vol. 166. Berlin: Springer, 1979
[5] Butkovic P, Hevery F. A condition for the strong regularity of matrices in the minimax algebra. Discrete Appl Math, 1985, 11: 209--222
[6] Butkovivc P, Cechlarova K, Szabc P. Strong linear independence in bottleneck algebra. Linear Algebra Appl, 1987, 94: 133--155
[7] Cechlarova K. Strong regularity of matrices in a discrete bottleneck algebra. Linear Algebra Appl, 1990, 128: 35--50
[8] Sanchez S. Resolution of composite fuzzy relation equations. Inform and Control, 1976, 30: 38--48
[9] Li Jian-Xin. The smallest solution of max-min fuzzy equations. Fuzzy Sets and Systems, 1990, 41: 317--327
[10] Guo S Z, Wang P Z, Di Nola A, Sesa S. Further contributions to the study of finite fuzzy relation equations. Fuzzy Sets and Systems, 1988, 26: 93--104
[11] Higashi M, Klir G J. Resolution of finite fuzzy relation equations. Fuzzy Sets and Systems, 1984, 13: 65--82
[12] Tan Y J. Eigenvalue and eigenvectors for matrices over distributive lattices. Linear Algebra Appl, 1998, 283: 257--272
[13] Tan Y J.On the eigenproblem of matrices over distributive lattices. Linear Algebra Appl, 2003, 374: 87--106
/
| 〈 |
|
〉 |