Left-continuous t-norms as functional algebras

With a left-continuous t-norm circle dot, we may associate the set of its vertical cuts, namely, the set F of functions f(a) : [0, 1] --> [0, 1], x --> x circle dot 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]; <= circle dot, 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 t-norms in a unified way. This subclass comprises most left-continuous t-norms discussed in the literature.