Characterizing dominates on a family of triangular norms

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 } _p=?∞ ^∞ of triangular norms given, for any real numberp ≠ 0, by $$T_p (a,b) = left[ {Max(a^p + b^p - 1,0} right]^{{1 mathord{left/ {vphantom {1 p}} right. kern-nulldelimiterspace} p}} $$ and, forp=?∞, 0 or ∞, by taking appropriate limits of those already defined. We sayT _q dominatesT _p provided $$T_q (T_p (a,b),T_p (c,d)) geqq T_p (T_q (a,c),T_q (b,d))$$ for alla,b,c,d in [0, 1]. The main result of this paper is that dominates is transitive on this family, in fact,T _q dominatesT _p if and only ifq ≦p.