Extendable local partial clones

We investigate interpolation and extrapolation properties of composition-closed sets of partial operations defined on an infinite set E. Considering local completeness (interpolation property) the structure of a maximal local partial clone is described via its intersections with the full partial clones on every finite k-element (k 2) subset of E. The criteria are established which characterize any finite domain partial operation that can be extended to an everywhere defined operation from the same local partial clone as well as the criteria describing a local partial clone (called extendable) in which every finite domain partial operation is extendable are given (extrapolation properties). Next the full list of partial orders on the countable set such that the partial clones of their partial n-endomorphisms are extendable is obtained. Finally, based on these criteria the description of all extendable maximal local partial clones defined on the countable set is provided. © 2007 Elsevier B.V. All rights reserved.