Resumen de Dominions of universal algebras and projective properties

A. I. Budkin

• Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ? A such that every pair of homomorphisms f, g: A ? ? {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group ?H, a? generated by one element modulo H.