Algebra & Number Theory

On coproducts in varieties, quasivarieties and prevarieties

George Bergman

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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, A0An, in a prevariety (S-closed class of algebras) P such that An generates P, and also subalgebras BiAi1 (0<in) such that for each i>0 the subalgebra of Ai1 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)iI such that the subalgebra of A generated by the Bi is their coproduct, and each Bi has a finite family of subalgebras (Cij)jJi 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 operation in P.

It is shown by example that for B a subgroup of the group S= Sym(Ω) of all permutations of an infinite set Ω, the group S need not have a subgroup isomorphic over B to the coproduct with amalgamation SBS. But under various additional hypotheses on B, the question remains open.

Article information

Algebra Number Theory, Volume 3, Number 8 (2009), 847-879.

Received: 10 June 2008
Revised: 23 November 2009
Accepted: 26 November 2009
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 08B25: Products, amalgamated products, and other kinds of limits and colimits [See also 18A30] 08B26: Subdirect products and subdirect irreducibility 08C15: Quasivarieties
Secondary: 03C05: Equational classes, universal algebra [See also 08Axx, 08Bxx, 18C05] 08A60: Unary algebras 08B20: Free algebras 20M30: Representation of semigroups; actions of semigroups on sets

coproduct of algebras in a variety or quasivariety or prevariety free algebra on $n$ generators containing a subalgebra free on more than $n$ generators amalgamation property number of algebras needed to generate a quasivariety or prevariety symmetric group on an infinite set


Bergman, George. On coproducts in varieties, quasivarieties and prevarieties. Algebra Number Theory 3 (2009), no. 8, 847--879. doi:10.2140/ant.2009.3.847.

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