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

Generalized fuzzy matrices are considered as matrices over a special type of semiring which is called an incline, and their transitive closure, powers, determinant and adjoint matrices are studied. An expression for the transitive closure of a matrix A as a sum of its powers and some sufficient conditions for powers of a matrix to converge are given. If the incline is commutative, a sufficient condition for nilpotency of a matrix is obtained, namely the determinants of the principal submatrices of the matrix are all equal to zero element. In addition, it is proved that A^(n-1) is equal to the adjoint matrix of A if the matrix A satisfies A>=In.