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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     author = {V. B. Alekseev},
     title = {On the cardinality of interval {Int(Pol}$_k$) in partial $k$-valued logic},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
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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/

[1] Alekseev V.B., Voronenko A.A., “O nekotorykh zamknutykh klassakh v chastichnoi dvuznachnoi logike”, Diskretn. matem., 6:4 (1994), 58–79 | Zbl

[2] Lau D., Function algebras on finite sets: a basic course on many-valued logic and clone theory, Springer Monographs in Mathematics, Springer, Berlin, 2006 | MR | Zbl

[3] Couceiro M., Haddad L., Schoelzel K., Waldhauser T., “A solution to a problem of D. Lau: Complete classification of intervals in the lattice of partial Boolean clones”, J. Mult.-Valued Logic Soft Comput., 28 (2017), 47–58 | MR | Zbl

[4] Dudakova O.S., “O klassakh chastichnykh monotonnykh funktsii shestiznachnoi logiki”, Problemy teoreticheskoi kibernetiki, Mat-ly XVIII Mezhdunar. konf. (Penza, 19–23 iyunya 2017 g.), ed. Yu.I. Zhuravlev, MAKS Press, M., 2017, 78–81

[5] Alekseev V.B., “O zamknutykh klassakh v chastichnoi $k$-znachnoi logike, soderzhaschikh klass monotonnykh funktsii”, Diskretn. matem., 30:2 (2018), 3–13

[6] Dudakova O.S., “Postroenie beskonechnogo semeistva klassov chastichnykh monotonnykh funktsii mnogoznachnoi logiki”, Vestn. Mosk. un-ta. Matem. Mekhan., 2019, no. 1, 3–7 | MR | Zbl

[7] Alekseev V.B., “O zamknutykh klassakh v chastichnoi $k$-znachnoi logike, soderzhaschikh vse polinomy”, Diskretn. matem., 33:2 (2021), 6–19 | Zbl

[8] Alekseev V.B., “On some intervals of partial clones”, J. Mult.-Valued Logic Soft Comput., 38 (2022), 3–22 | Zbl