aequationes mathematicae

, Volume 27, Issue 1, pp 255–273 | Cite as

Characterizing dominates on a family of triangular norms

  • Howard Sherwood
Research Papers

Abstract

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 ifqp.

AMS (1980) subject classification

Primary 26D20 Secondary 52A40 

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Copyright information

© Birkhäuser Verlag 1984

Authors and Affiliations

  • Howard Sherwood
    • 1
  1. 1.Department of StatisticsUniversity of Central FloridaOrlandoUSA

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