Clones closed with respect to permutation groups or transformation semigroups (Q1380842)

Let \(A\) be a finite set, \(S_A\) be the full symmetric group on \(A\), \(O_A^{(n)}\) be the set of all \(n\)-ary operations on \(A\), \(O_A= \bigcup_n O_A^{(n)}\), and \(\langle F\rangle\) be the clone generated by the set of operations \(F\subseteq O_A\). For a permutation \(s\in S_A\) and \(f\in O_A^{(n)}\), let \(f^s\) be defined by \(f^s(x_1, \dots, x_n)= (f(x_1^{s^{-1}}, \dots, x_n^{s^{-1}}))^s\). For a group \(G\subseteq S_A\) and a clone \(F\subseteq O_A\) define \(F^G= \{f^s \mid f\in F,\;s\in G\}\) and \(^GF= \langle F^G\rangle\). A clone \(F\) is \(G\)-closed (a \(G\)-clone), if \(^GF=F\), i.e. \(f^s\in F\) for all \(f\in F\), \(s\in G\). The set of all \(G\)-clones is a sublattice of the lattice \({\mathcal L}_A\) of all clones on \(A\); \(^GF\) is the least \(G\)-clone containing \(F\), \(_GF= \bigcap_{s\in G} F^s\) is the greatest \(G\)-clone contained in \(F\) and \(F\mapsto{^GF}\), \(F\mapsto{_GF}\) are closure operators on \({\mathcal L}_A\), which define corresponding equivalences on the lattice \({\mathcal L}_A\) and are used here to investigate this lattice. A general Galois theory of \(G\)-clones is outlined, some properties of the lattice of \(G\)-clones are described and generalisations considered, where instead of a permutation group \(G\) a transformation semigroup on \(A\) is used in the above definitions.