Geodesic
On the cardinality of interval Int(Pol$_k$) in partial $k$-valued logic
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2022), pp. 11-17 
V. B. Alekseev. On the cardinality of interval Int(Pol$_k$) in partial $k$-valued logic. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2022), pp. 11-17. http://geodesic.mathdoc.fr/item/VMUMM_2022_3_a2/
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     author = {V. B. Alekseev},
     title = {On the cardinality of interval {Int(Pol}$_k$) in partial $k$-valued logic},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {11--17},
     year = {2022},
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2022_3_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let Pol$_k$ be the set of all functions of $k$-valued logic representable by a polynomial modulo $k$, and let Int(Pol$_k$) be the family of all closed classes (with respect to superposition) in the partial $k$-valued logic containing Pol$_k$ and consisting only of functions extendable to some function from Pol$_k$. In this paper, we prove that if $k$ is divisible by the square of a prime number, then the family Int(Pol$_k$) contains an infinitely increasing (with respect to inclusion) chain of different closed classes. This result and the results obtained by the author earlier imply that the family Int(Pol$_k$) contains a finite number of closed classes if and only if $k$ is a prime number or a product of two different primes.

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