On Coproducts in Varieties, Quasivarieties and Prevarieties

If the free algebra F on one generator in a variety V of algebras (in the sense of universal algebra) has a subalgebra free on two generators, must it also have a subalgebra free on three generators? In general, no; but yes if F generates the variety V. Generalizing the argument, it is shown that if we are given an algebra and subalgebras, A0 ⊇ · · · ⊇ An, in a prevariety (S P -closed class of algebras) P such that An generates P, and also subalgebras Bi ⊆ Ai−1 (0 < i ≤ n) such that for each i > 0 the subalgebra of Ai−1 generated by Ai and Bi is their coproduct in P, then the subalgebra of A generated by B1, . . . , Bn is the coproduct in P of these algebras. Some further results on coproducts are noted: If P satisfies the amalgamation property, then one has the stronger “transitivity” statement, that if A has a finite family of subalgebras (Bi)i∈I such that the subalgebra of A generated by the Bi is their coproduct, and each Bi has a finite family of subalgebras (Cij)j∈Ji with the same property, then the subalgebra of A generated by all the Cij is their coproduct. For P a residually small prevariety or an arbitrary quasivariety, relationships are proved between the least number of algebras needed to generate P as a prevariety or quasivariety, and behavior of the coproduct