SOLIDIFYABLE MINIMAL CLONES OF PARTIAL OPERATIONS

Partial operations occur in the algebraic description of partial recursive functions and Turing machines (cf. A. I. Mal'cev⁸). Similarly to total operations superposition operations can also be defined on sets of partial operations. A clone of partial operations is a set of partial operations defined on the same base set A which is closed under superposition and contains all total projections. The collection of all clones of partial operations defined on a set A forms a complete lattice. For a finite nonempty set A this lattice is atomic and dually atomic. A partial algebra is said to be strongly solid if every strong identity of is satisfied as a strong hyperidentity in , i.e. if it is satisfied after any replacement of operation symbols by derived term operations of of the corresponding arity. A clone C of partial operations is called strongly solidifyable if there is a partial algebra such that C is equal to the clone of all term operations of . In this paper we determine all minimal strongly solidifyable clones of partial operations defined on a finite nonempty set A.