Left-continuous t-norms as Functional Algebras

With a left-continuous t-norm , we may associate the set of its vertical cuts, namely, the set F of functions fa : [0, 1] → [0, 1], x → x a. Endowed with the pointwise order, with the functional composition, with the constant 0 function and with the identity function, F is an algebra which is isomorphic to ([0, 1]; ≤, , 0, 1). We characterize the functional algebras arising in this way from left-continuous t-norms; the key property is that every two functions commute. On the basis of this approach, we describe a subclass of the left-continuous tnorms in a unified way. This subclass comprises most left-continuous t-norms discussed in the literature.

• Type: Conference
• Year: 2007
• Where: EUSFLAT
• Authors: Thomas Vetterlein
• Keywords: EUSFLAT 2007, Functions Fa, Left-continuous T-norms, Vertical Cuts