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Solution sets of \(\inf\)-\(\alpha_{\mathcal J}\) fuzzy relational equations on complete Brouwerian lattices. (English) Zbl 1129.03032
The study is concerned with the development of solutions of fuzzy relational equations defined on complete Brouwerian lattices. The equations under consideration assume the following form \[ \inf_{j\in{\mathbf J}}\alpha_F(a_j, x_j)= b, \] where \(a_j\) and \(b\) are elements of the complete Brouwerian lattice and \(\alpha_F\) is a general inverse operation of \(F\) with \(F\) belonging to a family of infinitely \(\vee\)-distributive pseudo-t-norms. The index \(j\) comes from some finite set \({\mathbf J}\). The elements \(a_j\) and \(b\) are given and \(x_j\) are to be determined. The equations under study form a generalization of well-known inf-\(\alpha\) fuzzy relational equations in case the pseudo-t norm is the minimum operation.
The study provides necessary and sufficient conditions required for the existence of a nonempty solution set. Furthermore it is shown how the maximal solutions to the equations are determined.

MSC:
03E72 Theory of fuzzy sets, etc.
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