R-implications and the exchange principle: The case of border continuous t-norms

Authors:
Balasubramaniam Jayaram;Michał BaczyńSki;Radko Mesiar
Affiliations:
Department of Mathematics, Indian Institute of Technology Hyderabad, Yeddumailaram - 502 205, India;Institute of Mathematics, University of Silesia, 40-007 Katowice, ul. Bankowa 14, Poland;Slovak University of Technology, Radlinskeho 11, 813 68 Bratislava, Slovakia and Ostrava University, IRAFM, 30. dubna 22, 701 03 Ostrava 1, Czech Republic
Venue:
Fuzzy Sets and Systems
Year:
2013

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Abstract

It is well-known that the residual of a left-continuous t-norm satisfies the exchange principle (EP). However, the left-continuity of a t-norm is only sufficient and not necessary, as many examples in the literature illustrate. In this work we study the necessary and sufficient conditions on a t-norm for its residual to satisfy (EP). We present a complete characterization of two classes of t-norms whose residuals satisfy (EP), viz., t-norms that are border-continuous and those that have an ordinal sum representation. Based on the obtained results we characterize t-norms, whose residuals satisfy both the exchange principle and the ordering property.