Characterizing dominates on a family of triangular norms (Q1079681)

Dominates is a relation which can be defined on any collection of operations which (1) are defined on the same partially ordered set and (2) have the same identity. In this paper the family considered is a family \(\{T_ p\}^{\infty}_{p=-\infty}\) of triangular norms given, for any real number \(p\neq 0\), by \(T_ p(a,b)=[Max(a^ p+b^ p- 1,0]^{1/p}\) and, for \(p=-\infty,0\) or \(\infty\), by taking appropriate limits of those already defined. We say \(T_ q\) dominates \(T_ p\) provided \[ T_ q(T_ p(a,b),T_ p(c,d))\geq T_ p(T_ q(a,c),T_ q(b,d)) \] for all a, b, c, d in [0,1]. The main result of this paper is that dominates is transitive on this family, in fact, \(T_ q\) dominates \(T_ p\) if and only if \(q\leq p\).