Monoids of Hypersubstitutions and M-solid Varieties
by
K. Denecke
University of Potsdam

The concept of a variety allows a systematic study of algebras and their interrelationships. The class of all algebras of a given type and, in particular, the class of all subvarieties of a given variety form complete lattices. These lattices are very complex and difficult to study. We want to describe a new method to study these lattices. This method is based on the concept of a hyperidentity which was introduced by W. Taylor in 1981. If every identity in a variety is satisfied as a hyperidentity, the variety is called solid. The collection of all solid varieties forms a complete sublattice of the lattice of all varieties of a given type of algebras. Using submonoids of the monoid of all hypersubstitutions one obtains complete lattices of M-solid varieties. This method is applicable for all types and all subvariety lattices, but in the several cases one has separate tools to get the concrete characterization of these complete lattices. We will present some results for varieties of type (2) and (2,2), especially for varieties of semigroups and semirings. Since this theory is based on a general theory of conjugate pairs of additive closure operators, it can be applied to classes different from varieties as quasivarieties and pseudovarieties. Clones closed under conjugation form one more application.

Date received: July 11, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caoc-31.