\(T\)-equalities as fuzzy non-archimedean metrics and \(\delta ^{+}\)-fuzzy equivalences of type \(S\)
The authors define a fuzzy tolerance relation \(E: X\times X\to [0,1]\) on a set J\(X\) pending the satisfaction of the following conditions: (i) \(E(x,x)= 1\), \(\forall x\in X\), (ii) \(E(x,y)= E(y,x)\), \(\forall x,y\in X\). Using the concept of t-norm, which is a mapping \(T: [0,1]\times [0,1]\to [0,1]\) satisfying the following axioms (i) \(T(a,1)= a\), \(\forall a\in I\), (ii) \(T(ab)= T(b,a)\forall a,b\in I\), (iii) \(a\leq c\), \(b\leq d\Rightarrow T(a,b)\leq T(c,d)\), (iv) \(T(a,T(b,c))= T(T(a, b),c)\forall a,b,c\in I\), the authors call a fuzzy tolerance relation \(E\) to be fuzzy \(T\)-equivalent on \(X\) if \(T(E(x, y), E(y,z))\leq E(x,x)\), \(\forall x,y,z\in X\). Fuzzy \(T\)-equality is obtained if another condition \(E(x,y)=1\Rightarrow x= y\) is satisfied. The authors then pass through a process of generalization using generalized distribution functions, define fuzzy \(S\)-metric space \((x,F)\), fuzzy \(S\)-Menger space \((X,F,T)\) relatively to \(T\), fuzzy equivalence of type \(S\) on \(X\), fuzzy equality of type \(S\) on \(X\), fuzzy \(T\)-equality of type \(S\) on \(X\), to finally reach the concept of fuzzy equivalence and deterministic distances.
scientific article; zbMATH DE number 5073286
scientific article