Some properties of infinite fuzzy relational equations on complete Brouwerian lattices.

*(English)*Zbl 1120.03041Summary: This paper deals with fuzzy relational equations \(A\odot X=b\) (where both \(A=(a_j)_{j\in J}\) and \(b\) are known, \(b\) has an irredundant continuous join-decomposition, \(X=(x_j)^T_{j\in J}\) is unknown, “\(\odot\)” represents “sup-inf” composition, \(J\) is an infinite index set) on complete Brouwerian lattices. At first, a necessary and sufficient condition for the 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.

##### MSC:

03E72 | Theory of fuzzy sets, etc. |

##### Keywords:

complete Brouwerian lattice; irredundant continuous join-decomposition; fuzzy relational equation; solution set
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\textit{X.-b. Qu} and \textit{X.-p. Wang}, Fuzzy Sets Syst. 158, No. 12, 1327--1339 (2007; Zbl 1120.03041)

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