On some geometric transformation of t-norms

Given a triangular norm $T$, its $t$-reverse $T^*$, introduced by C. Kimberling ({\it Publ. Math. Debrecen} 20, 21-39, 1973) under the name invert, is studied. The question under which conditions we have $ T^{**} = T$ is completely solved. The $t$-reverses of ordinal sums of $t$-norms are investigated and a complete description of continuous, self-reverse $t$-norms is given, leading to a new characterization of the continuous $t$-norms $T$ such that the function $ G(x,y) = x + y - T(x,y)$ is a $t$-conorm, a problem originally studied by M.J. Frank ({\it Aequationes Math.} 19, 194-226, 1979). Finally, some open problems are formulated