On the Cardinality of the Lattice of Closed Classes of Polynomials in 𝑘-Valued Logic for Composite Numbers 𝑘

<jats:p>Closed classes under superposition are examined in 𝑘-valued logic. E. Post established that the lattice (on inclusion) of all closed classes in two-valued logic is countable. Besides, each closed class has a finite basis in two-valued logic. Yu. I. Yanov and A. A. Muchnick proved that the lattice of all closed classes in 𝑘-valued logic is continuous at each 𝑘 &gt; 3. Besides, there are closed classes without a basis and closes classes of a countable basis in 𝑘-valued logic at 𝑘 &gt; 3. Because of the continuity on the lattice of all closed classes at 𝑘 &gt; 3, its sub-lattices are examined. In particular, the closed class of all functions, that are represented by polynomials modulo 𝑘, is considered in 𝑘-valued logic. This closed class contains all functions of 𝑘-valued logic, if and only if 𝑘 is a prime number. If 𝑘 is a composite number, then this closed class is not even pre-complete. In works of A. N. Cherepov, A. B. Remizov, A. A. Krokhin, K. L. Safin, E. V. Sukhanov, D. G. Meschaninov and of others the structure of sub-lattices and of over-lattices is examined for the closed class of all polynomial functions at composites 𝑘. In this work at each composite number 𝑘 the continuity of the sub-lattice is established for the closed class of all polynomial functions in 𝑘-valued logic.</jats:p>