Strict preference relations based on weak t-norms (Q1182218)

A concept of weak \(t\)-norm is introduced by removing from the number of conditions defining the notion of \(t\)-norm the commutativity and associativity conditions. The notion of right pseudocomplement of a weak \(t\)-norm plays also an important role. If \(w\) is a weak \(t\)-norm, \(w: I\times I\to I\) \((I=[0,1])\), then the function \(w^ \to: I\times I\to I\) is defined by \(w^ \to(a,b)=\sup\{x: w(a,x)\leq b\}\) and called the right pseudocomplement of \(w\). The concepts of weak \(t\)-norm and of its right pseudocomplement are used for general representation (via so called generator functions) of valued strict preference relations associated with a given valued preference relation. A value preference relation on \(A\) is simply a valued binary relation on \(A\), i.e. a function \(R: A\times A\to[0,1]\), where \(R(a,b)\) means the degree of preference of \(a\) over \(b\) when \(a,b\in A\). An antisymmetric valued relation \(P\) is called valued strict preference relation associated with the given valued preference relation \(R\) on \(A\) if \(P(a,b)=f(R(a,b),R(b,a))\), \(\forall a,b\in A\), where \(f\) is so called generator function that satisfies the following conditions: \(f: I\times I\to I\), \(f\) is nondecreasing with respect to the first argument and nonincreasing with respect to the second argument, \(x\leq y\) implies \(f(x,y)=0\). Among many interesting results the following theorem should be underlined. Assume that \(f\) is a generator function satisfying the following conditions: \(f(x,0)=x\), \(x\in I\), \(f(x,b)\) is left-continuous in \(x\) for \(b\in I\). In addition, let \(n\) be a given strict negation. Then there exists a weak \(t\)-norm \(w\) such that the following representation holds: \(f(x,y)=n[w^ \to(n^{- 1}(y),n^{-1}(x))]\). (A function \(n: I\to I\) is called strict negation if it is continuous, decreasing and \(n(0)=1\), \(n(1)=0\)).