Some Properties Of Transitive Matrices And Nilpotent Matrices On Lattices

In this paper, a binary operation is defined in a lattice, someproperties on the operation are introduced, such as monotonicity, associa-tivity and distributivity of the operation on operations∧and∨. Further,a binary operation is defined on matrices in a lattice. Some properties ofmatrices on the operation are examined, such as monotonicity, associativityand distributivity of operations and with respect to each other. Moreover,we prove that if R is a transitive matrix, then△R is transitive, nilpotentand irreflexive, show that if R is a nilpotent matrix, then (R/R)+ = R+, andexamine that if R is an irreflexive and transitive matrix, then R/R S Ris equivalent to R/R = S/R, so that we generalize the results obtained byHashimoto and Tan.