| In this paper, a fuzzy relational equation defined on completeBrouwer lattice is investigated. In particular, some properties of the minimalsolutions and maximal solution of the equation A☉X = b (where"☉"denotes sup - inf composition, A = (aj)j∈J, J is a finite set ) are given oncomplete Brouwerian lattice. Frist, we discuss the solution set in completeBrouwerian lattice, and give a method to calculate all of minimal solutionswhen the solution set is not empty. And then,the structure of solution set isdiscribed.Next, a necessary and su?cient condition describing the attainablesolution(resp.the unattainable solution)is proposed based on the coe?cientsof the equation .And when the solution set of the equation is not empty ,thestructure of the solution set is formulated.Further,We discuss some properties ofelements in nonnegative integer lattice and nonnegative integer lattice .Further,discribe the solution sets of @-Fuzzy relational equation on finite and infinitedomains.In the end ,some properties of idempotent matrixs are investigated.In particular,we discuss the decomposition problem of idempotent booleanmatrice, and give all its square roots.
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