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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 π‘˜-valued logic π‘ƒπ‘˜, then π‘†π‘‘π‘Ÿ(𝐴) is the set of all functions from the partial π‘˜-valued logic π‘ƒπ‘˜βˆ— , which can be expanded to a function from 𝐴. For any clone 𝐴 from π‘ƒπ‘˜, the set 𝐼𝑛𝑑(𝐴) of all partial clones in π‘ƒπ‘˜βˆ— lying between 𝐴 and π‘†π‘‘π‘Ÿ(𝐴) is investigated. We define a special family 𝑍(𝐴) of sets of predicates and prove that the lattice of partial clones in 𝐼𝑛𝑑(𝐴) (according to inclusion) is isomorphic to the lattice of sets in 𝑍(𝐴) (according to inclusion). For the set π½π‘˜ of all projections in π‘ƒπ‘˜, we prove that the cardinality of 𝐼𝑛𝑑(π½π‘˜) is continuum. For the set π‘ƒπ‘œπ‘™π‘˜ of all polynomials in π‘ƒπ‘˜ where π‘˜ is a product of two different prime numbers, we prove that 𝐼𝑛𝑑(π‘ƒπ‘œπ‘™π‘˜) consists of 7 partial clones which are completely described.

Keywords: π‘˜-valued logic, partial π‘˜-valued logic, clone, partial clone, predicate, projection, polynomial

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