Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms

Several properties of quasi- and pseudo-inverses of a non-decreasing real function are discussed. Based on a result of Schweizer and Sklar, for a given triangular norm T and non-decreasing function f a construction method leading to a commutative, fully ordered semigroup on the unit interval is given. A similar construction based on the pseudo-inverse implies that the resulting operation will be bounded from above by the minimum, but then the associativity may be violated. Several sufficient conditions for constructing new t-norms from a given one and a non-decreasing function f, based on its quasi-inverses and on its pseudo-inverse, respectively, are discussed, together with illustrative examples. (C) 1999 Published by Elsevier Science B.V. All rights reserved.