Some Results Of The Solution Set Of Infinite Fuzzy Relational Equations On Complete Brouwerian Lattices

In this paper, an infinite fuzzy relational equation defined oncomplete Brouwerian lattice is investigated. In particular, some properties ofthe solution set of the equation A (?) X = b (where"(?)"denotes sup-productcomposition) are given when the domain is an infinite set on lattice [0,1]. Fromthe coeffcients of the equation A (?) X = b, it is showed that a su?cient andnecessary condition that the solution set is nonempty and one for existence of anattainable solution or an unattainable solution. Further, when the solution setis nonempty, the structure of the solution set of the equation is given. Next, theproperties of the solution set of the fuzzy relational equation sup_j∈JT (a_j,x_j) = b(where T is a pseudo t-norm) are given when the domain is an infinite seton complete distributive lattice L. When b is a continuous join-irreducibleelement, a suffcient and necessary condition describing the attainable solution(resp. the unattainable solution) is formulated,and some properties of theattainable solution (resp. the unattainable solution) are shown. Further, whenthe solution set is nonempty, the structure of the solution set of the equation isgiven. In the end, the solution set of equation A@X = B(where"@"denotesinf-αcomposition, A = (a_ij)I×J,B = (bi_{i∈I}^T) defined on [0,1] is investigated.When the domain I is finite, J is infinite, suffcient and necessary conditions forexistences of an attainable solution and a maximal solution are obtained.