Compositions of fuzzy relations based on aggregation operator and its pseudo-complement (Q2747058)

The authors deal with some generalizations of composition of two fuzzy relations. To this end they consider as aggregation operator a binary operation \(F\) in \([0,1]\) such that \(F\) is non-decreasing in each place, \(F(0,0)= 0\), \(F(1,1)= 1\), and \(F\) is commutative. When \(F\) is semicontinuous, its pseudocomplement \(f\) is given by \(afb= \sup\{x[0,1]|F(a,x)\leq b\}\). Then, the following compositions of two fuzzy relations \(P\), \(Q\) are defined: NEWLINE\[NEWLINE\Biggl(P \underset\vee{}\@ Q\Biggr)(x,y)= \bigvee_{y\in Y} F(P(x,y), Q(y,z)),\;\Biggl(P \underset\wedge{}\@ Q\Biggr)(x,z)= \bigwedge_{y\in Y} (P(x,y)fQ(y,z)).NEWLINE\]NEWLINE Basic properties are given.