The Dimension of a Kernel and the Semantical Kernel of a Hypersubstitution
by
K. Denecke
University of Potsdam
Coauthors: Shelly L. Wismath

In [Grac-S;07] the authors introduced the dimension dim(V) of a variety V as cardinality of the set of all proper derived varieties which are included in V. We show that dim(V) is less or equal to the cardinal of the natural numbers and give an example of a variety V where the dimension is the cardinal of the natural numbers. For every non-negative integer n there is a variety V such that dim(V) = n. In [Den-K-N;02] the authors introduced the concept of a semantical kernel of a hypersubstitution. We show that dim(V) is the cardinality of the set of all semantical kernels of all hypersubstitutions with respect to V. Using this connection and some results from [Den-K-N;02] we determine the dimension of the variety Alg(n) of all algebras of type n, where n is greater or equal to 1. Moreover, for some small non-negative integers we characterize all solid varieties which have this integer as its dimension.

[Grac-S;07] E. Grazcynska, D. Schweigert, The Dimension of a Variety, Discussiones Mathematicae, General Algebra and Applications 27 (2007), 35-47.

[Den-K-N;02] Denecke, K., Koppitz, J., Niwczyk, St., Equational theories generated by generalized hypersubstitutions of type (n), Int. Journal of Algebra and Computation, Vol. 12, No. 6 (2002), 867-876.

Date received: April 18, 2008