A semigroup of operators in convexity theory (Q2781397)

The important role in convex analysis of the following three operators is well-known: (1) the convex hull \(c(f)(x)\), \(x\in E\), \(f\colon E\to\mathbb{R}\cup\pm\infty\), \(E\) a real vector space; (2) the largest lower semicontinuous minorant \(l(f)(x)\); (3) \(m(f)(x)=f(x)\), if \((\forall x) f(x)>-\infty\), or \(m(f)(x)\equiv-\infty\), if \((\exists x) f(x)=-\infty\). This work is an exhaustive investigation of the semigroup generated by the identity and the operators \(c,l,m\). The author answers the questions (from the introduction): How many elements are there in this semigroup? What structure does it have with respect to composition and with respect to the natural order? Is it a lattice under this order? Is there a representation of the semigroup as a semigroup of matrices?