On the number of closed subsets of linear functions in the 6-valued logic (Q810496)

Let \(E_ k:=\{0,1,...,k-1\}\), \(P_ k^{(n)}:=\{f^{(n)}|\) \(f^{(n)}: E_ k^ n\to E_ k\}\), and let \(P_ k:=\cup_{n\geq 1}P_ k^{(n)}\) be the set of all functions of the k-valued logic. With \(\zeta\), \(\tau\), \(\Delta\), \(\nabla\) and * we denote the Mal'tsev operations over \(P_ k\). Then the iterative algebra \({\mathfrak P}_ k:=(P_ k;\zeta,\tau,\Delta,\nabla,*)\) is an algebra of the type (1,1,1,1,2). A subset of \(P_ k\) is said to be closed if it is the underlying set of a subalgebra of \({\mathfrak P}_ k\). One of the well examined closed subsets of \(P_ k\) is the set \[ L_ k:=\cup_{n\geq 1}\{f^{(n)}\in P_ k| \quad \exists a_ 0,...,a_ n\in E_ k:\;f(x_ 1,...,x_ n)=a_ 0+\sum^{n}_{n=1}a_ i\cdot x_ i mod k\}. \] In particular, A. A. Salomaa, J. Bagyinszkij and J. Demetrovics determined the lattice \({\mathcal L}_ k\) of all closed subsets of \(L_ k\) for the case that k is a prime number. By A. Szendrei it was proved that all closed sets of linear functions in square-free-valued logic are finitely generated and the order of these sets is at most 3. Furthermore, it is known that \(| {\mathcal L}_ k| =\aleph_ 0\) if k is not square-free. The main purpose of the present note is to prove that the number of closed subsets of \(L_ 6\) is 7524. We remark that the following table is a consequence of this: \[ \begin{matrix} k& 2& 3& 4& 5& 6& 7& 8& 9\\ |{\mathcal L}_ k| & 15& 38& \aleph_ 0& 319& 7524& 470& \aleph_ 0& \aleph_ 0 \end{matrix} \]