On Prepolynomially Complete Algebras
Dietmar Schweigert · Journal of the London Mathematical Society · 1979
In the theory of multivalued logic I. G. Rosenberg has found a fundamental theorem characterizing functional completeness by relations. In this approach the precomplete classes play an essential role. The aim of this paper is to study instead of the clones of functions of a multivalued logic the clones of polynomial functions of algebras with a carrier set of a fixed finite cardinality. Concerning the concept of functional completeness we are considering polynomially complete algebras and concerning the precomplete classes we introduce the concept of prepolynomially complete algebras. We can * restate the theorem of I. G. Rosenberg and have as a result that the finite algebras fall into the classes of five types 0, L, C, Z, R and the class of polynomially complete algebras. We give some examples for these classes confining ourselves to lattices, groups and vector spaces. 1. Clones If [I is a m-place function and A is defined by This (m + l)-place operation is called (m, rc)-composition.