On a semigroup of Marcinkiewicz modulars with involution

The set $\mathbf{M}$ of all concave Marcinkiewicz modulars on $[0,1]$ is a semigroup with respect to the usual composition of functions. It is established that some properties of modulars (which are of importance in interpolation and in general Banach space theory) distinguish subsets of $\mathbf{M}$ that form ideals of the semigroup. These ideals turn out to be in a natural duality relation, which is also studied.