On Some Intervals of Partial Clones
Valeriy B. Alekseev
This paper deals with clones, i.e. sets of functions containing all projections and closed under compositions. If 𝐴 is any clone from the 𝑘-valued logic 𝑃𝑘, then 𝑆𝑡𝑟(𝐴) is the set of all functions from the partial 𝑘-valued logic 𝑃𝑘∗ , which can be expanded to a function from 𝐴. For any clone 𝐴 from 𝑃𝑘, the set 𝐼𝑛𝑡(𝐴) of all partial clones in 𝑃𝑘∗ lying between 𝐴 and 𝑆𝑡𝑟(𝐴) is investigated. We define a special family 𝑍(𝐴) of sets of predicates and prove that the lattice of partial clones in 𝐼𝑛𝑡(𝐴) (according to inclusion) is isomorphic to the lattice of sets in 𝑍(𝐴) (according to inclusion). For the set 𝐽𝑘 of all projections in 𝑃𝑘, we prove that the cardinality of 𝐼𝑛𝑡(𝐽𝑘) is continuum. For the set 𝑃𝑜𝑙𝑘 of all polynomials in 𝑃𝑘 where 𝑘 is a product of two different prime numbers, we prove that 𝐼𝑛𝑡(𝑃𝑜𝑙𝑘) consists of 7 partial clones which are completely described.
Keywords: 𝑘-valued logic, partial 𝑘-valued logic, clone, partial clone, predicate, projection, polynomial