## Algebra & Number Theory

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

### On coproducts in varieties, quasivarieties and prevarieties

#### 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, ${A}_{0}\supseteq \cdots \supseteq {A}_{n}$, in a prevariety ($\mathbb{S}\mathbb{P}$-closed class of algebras) $P$ such that ${A}_{n}$ generates $P$, and also subalgebras ${B}_{i}\subseteq {A}_{i-1}$ $\left(0<i\le n\right)$ 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 ${B}_{1},\dots ,{B}_{n}$ 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 ${\left({B}_{i}\right)}_{i\in I}$ such that the subalgebra of $A$ generated by the ${B}_{i}$ is their coproduct, and each ${B}_{i}$ has a finite family of subalgebras ${\left({C}_{ij}\right)}_{j\in {J}_{i}}$ 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 $S=Sym\left(\Omega \right)$ of all permutations of an infinite set $\Omega $, the group $S$ need not have a subgroup isomorphic over $B$ to the coproduct with amalgamation $S\phantom{\rule{0.3em}{0ex}}{\coprod}_{B}S$. 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**

Received: 10 June 2008

Revised: 23 November 2009

Accepted: 26 November 2009

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

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

**Subjects**

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

**Keywords**

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

#### 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