Clones with Nullary Operations

This article discusses clones with nullary operations and the corresponding relational clones, both defined on arbitrary sets. By means of two pairs of kernel and closure operators, the relationship between such clones and clones in the traditional sense, i.e. without nullary operations, is investigated, and in particular the latter type of clones is located in the lattice of all clones. Finally, the fundamental theorem characterising Galois closed sets w.r.t. an adjusted version of Pol – Inv as local closures of clones and relational clones, respectively, is proven in the more comprehensive setting.