On categorical equivalence of finite arithmetical algebras
by
Kalle Kaarli
University of Tartu, Estonia

C. Bergman [1] completely solved the categorical equivalence problem for finite algebras with majority term. As a special case, he proved that the categorical equivalence class of a finite algebra A which has no proper subalgebras and generates an arithmetical variety, is fully determined by the full subcategory Q(A) of V(A) whose objects are the quotient algebras of A. The present work addresses the question: what is the formal structure of the categories Q(A) which appear in this situation? So far we have mainly studied algebras with all quotients congruence rigid. The latter means that every automorphism preserves all congruences. The system of automorphism groups of the quotients of A is a main example of what we call a group scheme. If all quotients of A are congruence rigid then the group scheme of A determines the category Q(A). We study which group schemes give raise to finite algebras having no proper subalgebras and generating arithmetical varieties.

References 1. C. Bergman, Categorical equivalence of algebras with majority term. Algebra Universalis 40 (1998), 149-175.

Date received: December 31, 2001

Copyright © 2001 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 # caig-30.