On the Cardinality of Interval Int(Pol$${}_{\boldsymbol{k}}$$) in Partial $$\boldsymbol{k}$$-Valued Logic
Valeriy B. Alekseev · Moscow University Mathematics Bulletin · 2022
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.