On coproducts in varieties, quasivarieties and prevarieties

TL;DR

This paper investigates the properties of coproducts in varieties, quasivarieties, and prevarieties, providing conditions under which certain subalgebras form coproducts and exploring implications for algebraic structures like groups.

Contribution

It generalizes coproduct construction conditions in prevarieties and quasivarieties, and examines their behavior under various algebraic properties and assumptions.

Findings

Subalgebra free on two generators implies free on three if F generates the variety.

Coproducts satisfy a transitivity property under the amalgamation property.

Relationships between generating sets and coproduct behavior in quasivarieties are established.

Abstract

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, A_0\supseteq ... \supseteq A_n, in a prevariety (SP-closed class of algebras) P such that A_n generates P, and also subalgebras B_i\subseteq A_{i-1} (0<i\leq n) such that for each i>0 the subalgebra of A_{i-1} generated by A_i and B_i is their coproduct in P, then the subalgebra of A generated by B_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 the subalgebra of A generated by the B_i is their coproduct, and each B_i has a finite family of subalgebras (C_{ij})_{j\in J_i} with the same property, then the 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 to generate P as a prevariety or quasivariety, and behavior of the coproduct operation in P. It is shown by example that for B a subgroup of the group S = Sym(\Omega) of all permutations of an infinite set \Omega, the group S need not have a subgroup isomorphic over B to the coproduct with amalgamation S \cP_B S. But under weak additional hypotheses on B, the question remains open.