Some properties of infinite fuzzy relational equations on complete Brouwerian lattices

This paper deals with fuzzy relational equations A circle dot X = b (where both A = (a(j)) (j) (is an element of j) and b are known, b has an irredundant continuous join-decomposition, X = (x(j))(T) (j) (is an element of) (J) is unknown, "circle dot" represents "sup-inf" composition, J is an infinite index set) on complete Brouwerian lattices. At first, a necessary and sufficient condition for existence of the attainable solutions (the unattainable solutions, respectively) and some properties of the attainable solutions (the unattainable solutions, respectively) are formulated. Then the solution set of the fuzzy relational equations is described. (c) 2007 Elsevier B.V. All rights reserved.