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

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.