Generators, indecomposables and free algebras (Q1119674)

Let \({\mathfrak A}=(A,\tau)\) be an algebra, \(\tau\) being the set of operation symbols of finite arities defined on A. Denote by \(T^{\alpha}\) the algebra of the \(\alpha\)-ary terms, where \(\alpha\) is a cardinal. It is known that \({\mathfrak A}\) is generated by n of its elements if and only if it is a homomorphic image of \(T^ n\). This is equivalent with ``having no more than n indecomposables'' in positive first-order finitary logic. The author proves the following theorem: Let \({\mathfrak A}\) be a \(\tau\)- algebra and n a natural number which is different from zero if \(\tau\) does not have constants. If \(\tau\) is infinite, \({\mathfrak A}\) is the homomorphic image of an elementary extension of \(T^ n\). If \(\tau\) is finite, this occurs precisely when \({\mathfrak A}\) has no more than n indecomposables.