Method of solution to fuzzy relation equations in a complete Brouwerian lattice (Q5939642)

The author studies fuzzy relation equations defined in complete Brouwerian lattices \((L)\) of the form NEWLINE\[NEWLINEb=A\circ XNEWLINE\]NEWLINE viz. NEWLINE\[NEWLINEb= \bigvee^n_{i=1} (a_i\wedge x_i)NEWLINE\]NEWLINE where \(b\in L\), \(A=[a_1 a_2 \dots a_n]\), \(X=[x_1 x_2 \dots x_n]^T\) while \(A\) and \(X\) are convoluted via a max-min composition \((\vee-\wedge)\).NEWLINENEWLINENEWLINEThe main result is about a minimal solution to the above equation. Given are two sufficient and necessary conditions under which this minimal solution exists (assuming that there is a nonempty solution to the equation set and \(b\) comes with an irredundant finite join-decomposition). An illustrative numerical example is included.