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The transitive closure, convergence of powers and adjoint of generalized fuzzy matrices. (English) Zbl 1068.15022

Matrices over “soft” algebraic structures were considered many years ago, e.g.: Boolean matrices [R. D. Luce, Proc. Am. Math. Soc. 3, 382–388 (1952; Zbl 0048.02302)], matrices over lattices [Y. Give’on, Inf. Control 7, 477–484 (1964; Zbl 0154.01103)] or matrices over residuated groupoids [T. S. Blyth, J. Lond. Math. Soc. 39, 427–432 (1964; Zbl 0154.01104)].
This paper deals with matrices over a semiring \((R,\vee,\star)\), where \((R,\vee,0,1)\) is a bounded semilattice and \((R,\star,1)\) is a monoid with operation \(\star\) distributive over \(\vee\). Such algebraic structure was introduced by M. Yoeli [Am. Math. Mon. 68, 552–557 (1961; Zbl 0115.02103)] as “Q-semiring”. Now it is called “incline” after Z.-Q. Cao, K. H. Kim and F. W. Roush [Incline algebra and applications, Wiley, New York (1984; Zbl 0541.06009)]. The paper generalizes some results on the title matrix notions known before for: Boolean matrices [K. H. Kim, Boolean matrix theory and applications, Dekker, New York (1982; Zbl 0495.15003)], fuzzy matrices [M. G. Thomason, J. Math. Anal. Appl. 57, 476–480 (1977; Zbl 0345.15007)] and matrices over lattices [Y.-J. Tan, Linear Algebra Appl. 386, 1–14 (2001)].

MSC:

15B33 Matrices over special rings (quaternions, finite fields, etc.)
16Y60 Semirings
06F05 Ordered semigroups and monoids
08A72 Fuzzy algebraic structures
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References:

[1] Z.Q. Cao, K.H. Kim, F.W. Roush, Incline Algebra and Applications, Ellis Horwood, Chichester, England, Wiley, New York, 1984.; Z.Q. Cao, K.H. Kim, F.W. Roush, Incline Algebra and Applications, Ellis Horwood, Chichester, England, Wiley, New York, 1984. · Zbl 0541.06009
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[10] Tan, Y., On the powers of matrices over a distributive lattice, Linear Algebra Appl., 336, 1-14 (2001) · Zbl 0992.15017
[11] Thomason, M. G., Convergence of powers of a fuzzy matrix, J. Math. Anal. Appl., 57, 476-480 (1977) · Zbl 0345.15007
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[13] Zhang, K. L., On the nilpotent matrices over \(D_{01}\)-lattice, Fuzzy Sets and Systems, 117, 403-406 (2001) · Zbl 0971.15008
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