Large k-free algebras (Q1078593)

Let \({\mathcal V}\) be a variety of universal algebras. We say that an algebra A in \({\mathcal V}\) is k-free iff there are \(x_ 1,...,x_ k\) in A with the property that for any \(y_ 1,...,y_ k\) in A, there is exactly one endomorphism \(f: A\to A\) with \(f(x_ i)=y_ i\), \(1\leq i\leq k\), (thus ''0-free'' means the same as ''rigid'', i.e. an algebra with no non-trivial endomorphism). The authors prove the conjecture that in a category with free objects and arbitrarily large rigid objects, in particular in the varieties with large rigid objects, there will be also arbitrarily large k-tree ones. They first prove that in some special cases (semigroups, commutative groupoids) the sum construction works, after having shown with an opportune counterexample that the sum idea does not always work they give another construction.