On some algebraic and topological properties of generated border-continuous triangular norms

In the class of all border-continuous triangular norms generated by strictly decreasing additive generators the following algebraic and topological properties are studied in detail: the continuity (left-continuity/right-continuity), the border-continuity, the conditional cancellation law, the cancellation law, the Archimedean Property, the diagonal property, the continuity on the diagonal and also the associative law. The relations among these properties are examined. A special attention is devoted to the set of all idempotent elements of generated functions and to the set of all points of discontinuity of an additive generator. The necessary and sufficient conditions for a generated function to be a border-continuous triangular norm are expressed in terms of properties of generated functions. Some relevant examples and counterexamples are indicated.