The Dimension of a Variety and the Kernel of a Hypersubstitution

Abstract

The dimension of a variety V of algebras of a given type was introduced by E. Graczyńska and D. Schweigert in [7] as the cardinality of the set of all derived varieties of V which are properly contained in V . In this paper, we characterize all solid varieties of dimensions 0, 1, and 2; prove that the dimension of a variety of finite type is at most ℵ 0 ; give an example of a variety which has infinite dimension; and show that for every n ∈ N there is a variety with dimension n. Finally, we show that the dimension of a variety is related to the concept of the semantical kernel of a hypersubstitution and apply this connection to calculate the dimension of the class of all algebras of type τ = (n).