The structure of the lattice of closed classes of polynomials

In this article the structure of the lattice of closed classes of polynomials modulo $k$ is investigated. More precisely, we investigate the structure of the interval of this lattice from the class of all linear polynomials with zero constant term to the class of all polynomials modulo $k$. It is proved that this interval (as partially ordered set) is the direct product of two subintervals, and its structure is completely determined when $k$ is square free. Moreover, for $k=4$ (minimal not square free $k$) the description of the interval from the class of all linear polynomials to the class of all polynomials is given.