A note on homomorphic images, subalgebras and various products

Let I, H, S, P, P_fin, P_f , P_u, P_s be the usual operators on classes of algebras of the same type (P, P_fin, P_f, P_u, P_s are respectively for direct, finite, filtered, ultra and subdirect products). The partially ordered monoid generated by the operators H, S, P with respect to composition of operators, I as an identity element, and natural ordering between operators is described by Pigozzi (Algebra Universalis 2 (1972), 346–353). The aim of this note is to describe the partially ordered monoids generated by H, S, P_u and by H, S, P_s and as well to summarize known results on the partially ordered monoids of operators generated by H, S and some of the above introduced products.Dedicated to the memory of Aleksandar PopoviReceived April 1, 2001; accepted in final form July 11, 2004.