| In this paper, a fuzzy relational equation defined on complete Brouwerian lattice is investigated. In particular, some properties of the solution set of the equation A 0 X = b (where "0" denotes sup-inf composition) are given when the domain is an infinite set on lattice [0, 1]. Prom the coefficients of the equation AQX = 6, it is showed that a sufficient and necessary condition that the solution set is nonempty and one for existence of an attainable solution or an unattainable solution. Further, when the solution set is nonempty, the structure of the solution set of the equation is given. Next, the solution set of the equation AoX = B (where "o" denotes sup-product composition, A = (a,ij)I J, B = (bi)T ) defined [0, 1] is investigated. When the domain / is finite, J is infinite, sufficient and necessary conditions for existences of an attainable solution and a minimal solution are obtained. Next, the properties of the solution set of the fuzzy relational equation sup T(ai,xi) = b (where T is a pseudo i-norm) on complete lattice L are discussed. When the domain is a finite set, b is an irreducible element or b has an irredundant finite join-decomposition in the case that / is finite, the solution set of the equation sup T(ai,xi) = b is showed. When / is infinite and 6 is a compact element and with an irredundant finite join-decomposition, it is showed that a sufficient and necessary condition that the solution set is nonempty. It is also proved that there exists a minimal solution if the solution set of the equation sup T(ai,xi) = b is nonempty. In the end, when the domain is finite, b is a meet-irreducible element or 6 has an irredundant finite meet-decomposition, a sufficient and necessary condition that the solution set is nonempty is given and the number of distinct minimal solutions is formulated. Further, it is proved that there exists a minimal solution such that for every if the solution set .
|
PDF Full Text Request