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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 π΄ is any clone from the π-valued logic ππ, then ππ‘π(π΄) is the set of all functions from the partial π-valued logic ππβ , which can be expanded to a function from π΄. For any clone π΄ from ππ, the set πΌππ‘(π΄) of all partial clones in ππβ lying between π΄ and ππ‘π(π΄) is investigated. We define a special family π(π΄) of sets of predicates and prove that the lattice of partial clones in πΌππ‘(π΄) (according to inclusion) is isomorphic to the lattice of sets in π(π΄) (according to inclusion). For the set π½π of all projections in ππ, we prove that the cardinality of πΌππ‘(π½π) is continuum. For the set ππππ of all polynomials in ππ where π is a product of two different prime numbers, we prove that πΌππ‘(πππ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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