A note on "decomposition problem of fuzzy relations"

The decomposition problem of a fuzzy relation R is an element of F(X x X) c an be stated as: "Given a fuzzy relation R is an element of F(XxX), to dete rmine whether there exists a fuzzy relation Z is an element of F(X x X) suc h that R = Z.Z, where X is a finite set and "." is the max-min composition of two fuzzy relations." In particular, if R is a Boolean matrix, then this problem becomes to find the square root of a Boolean matrix, which is a we ll-known unsolved problem. In 1985, Di Nola et al. (A. Di Nola, S. Sessa an d W. Pedrycz, Int. J. General Systems, 10, 1985, 123-133) had solved it in theory, and proposed a numerical algorithm, illustrated by a flowchart. In this note, we first point out that the flowchart proposed by Di Nola er al. is in error and give a correct flowchart. Then we give a numerical example , which is also a counterexample of the flowchart given by Di Nola et al., to explain our flowchart.