The transitive closure, convergence of powers and adjoint of generalized fuzzy matrices (Q1885728)

Matrices over ``soft'' algebraic structures were considered many years ago, e.g.: Boolean matrices [\textit{R. D. Luce}, Proc. Am. Math. Soc. 3, 382--388 (1952; Zbl 0048.02302)], matrices over lattices [\textit{Y. Give'on}, Inf. Control 7, 477--484 (1964; Zbl 0154.01103)] or matrices over residuated groupoids [\textit{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 \textit{M. Yoeli} [Am. Math. Mon. 68, 552--557 (1961; Zbl 0115.02103)] as ``Q-semiring''. Now it is called ``incline'' after \textit{Z.-Q. Cao, K. H. Kim} and \textit{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 [\textit{K. H. Kim}, Boolean matrix theory and applications, Dekker, New York (1982; Zbl 0495.15003)], fuzzy matrices [\textit{M. G. Thomason}, J. Math. Anal. Appl. 57, 476--480 (1977; Zbl 0345.15007)] and matrices over lattices [\textit{Y.-J. Tan}, Linear Algebra Appl. 386, 1--14 (2001)].