Coproducts in varieties of affine modules
by
M. Stronkowski
Warsaw University of Technology, Poland

Any variety of algebras can be considered as a category with algebras as objects and homomorphisms as morphisms. Such categories have coproducts and we can ask about their structure. The answer to this question is relevant for a description of free algebras in a variety V. It is well-known that the coproduct of family of free V-algebras X_iV over sets X_i is a free V-algebra over the disjoint union of the sets X_i. Thus knowledge of "small" free algebras, for instance over one or two free generators, and a good structural description of coproducts, can provide a good description of "larger" free V-algebra. This type of consideration was undertaken by Bela Csákány [1], who has shown that under certain general conditions, the coproduct of any two algebras in a given variety V coincides with their product iff V is equivalent to a variety of semimodules. In this note we are interested in varieties in which the coproduct of any two algebras A and B is isomorphic to A ×B ×2V, where 2V is a free V-algebra over two free generators. Such types of coproducts characterize varieties of affine modules. We show that a variety V has coproducts of this type iff it is equivalent to a variety of affine modules. As corollaries we obtain the known characterizations of such varieties as idempotent central Mal'cev varieties (see [3]) and as idempotent hamiltonian regular varieties (see [2]).

  [1] B. Csákány, Varieties equivalent to varieties of semimodules and modules, Acta Sci. Math. 24 (1963) 157-164, in Russian.

[2] B. Csákány, Varieties of affine modules, Acta Sci. Math. 37 (1973) 3-10.

[3] A.B. Romanowska and J.D.H. Smith, Modes, World Scientific, Singapore 2002.

Date received: January 27, 2003