Morita equivalence of many-sorted algebraic theories

Algebraic theories are called Morita equivalent provided that the corresponding varieties of algebras are equivalent. Generalizing Dukarm’s result from one-sorted theories to many-sorted ones, we prove that all theories Morita equivalent to an S-sorted theory T are obtained as idempotent modifications of T . This is analogous to the classical result of Morita that all rings Morita equivalent to a ring R are obtained as idempotent modifications of matrix rings of R. © 2006 Published by Elsevier Inc. * Corresponding author. E-mail address: [email protected] (J. Adámek). 1 Research supported by the Czech Grant Agency, Project 201/02/0148. 2 Partial financial assistance by Centro de Matemática da Universidade de Coimbra/FCT. 3 Partial financial assistance by Centro de Matemática da Universidade de Coimbra/FCT and Escola Superior de Tecnologia de Viseu. 0021-8693/$ – see front matter © 2006 Published by Elsevier Inc. doi:10.1016/j.jalgebra.2006.01.014 362 J. Adámek et al. / Journal of Algebra 297 (2006) 361–371