## Algebra & Number Theory

### On coproducts in varieties, quasivarieties and prevarieties

George Bergman

#### 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, $A0⊇⋯⊇An$, in a prevariety ($Sℙ$-closed class of algebras) $P$ such that $An$ generates $P$, and also subalgebras $Bi⊆Ai−1$ $(0 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 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 $S∐BS$. But under various additional hypotheses on $B$, the question remains open.

#### Article information

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

Dates
Revised: 23 November 2009
Accepted: 26 November 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797499

Digital Object Identifier
doi:10.2140/ant.2009.3.847

Mathematical Reviews number (MathSciNet)
MR2587406

Zentralblatt MATH identifier
1194.08002

#### Citation

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. https://projecteuclid.org/euclid.ant/1513797499