One-point extension of the algebra of incompletely specified operations

Abstract

Incompletely specified operations on a finite set are operations with values specified only for some elements of the domain. The set of all such mappings, together with naturally introduced fundamental operations, forms the algebra of incompletely specified operations. On a twoelement set, it is isomorphic to the full algebra of hyperoperations. On a set with at least three elements, there is no suitable homomorphism to algebras of total, partial or hyperoperations. An incompletely specified operation on a set induces an operation on the set extended with one additional element. We consider the full algebra of those extended incompletely specified operations. In general, sets of extended incompletely specified operations that preserve a given relation are not closed under fundamental operations of the algebra. However, certain classes of relations possess this property and two of them are presented in this paper. © 2014 Old City Publishing, Inc.