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If the free algebra F on one generator in a variety V of algebras (in thesense of universal algebra) has a subalgebra free on two generators, must italso have a subalgebra free on three generators? In general, no; but yes if Fgenerates the variety V.  Generalizing the argument, it is shown that if we are given an algebra andsubalgebras, A_0\supseteq ... \supseteq A_n, in a prevariety (SP-closed classof algebras) P such that A_n generates P, and also subalgebras B_i\subseteqA_{i-1} (0<i\leq n) such that for each i>0 the subalgebra of A_{i-1} generatedby A_i and B_i is their coproduct in P, then the subalgebra of A generated byB_1, ..., B_n is the coproduct in P of these algebras.  Some further results on coproducts are noted:  If P satisfies the amalgamation property, one has the stronger "transitivity"statement: if A has a finite family of subalgebras (B_i)_{i\in I} such that thesubalgebra of A generated by the B_i is their coproduct, and each B_i has afinite family of subalgebras (C_{ij})_{j\in J_i} with the same property, thenthe subalgebra of A generated by all the C_{ij} is their coproduct.  For P a residually small prevariety or an arbitrary quasivariety,relationships are proved between the least number of algebras needed togenerate P as a prevariety or quasivariety, and behavior of the coproductoperation in P.  It is shown by example that for B a subgroup of the group S = Sym(\Omega) ofall permutations of an infinite set \Omega, the group S need not have asubgroup isomorphic over B to the coproduct with amalgamation S \cP_B S. Butunder weak additional hypotheses on B, the question remains open.

On coproducts in varieties, quasivarieties and prevarieties