Algebra & Number Theory
- Algebra Number Theory
- Volume 3, Number 8 (2009), 847-879.
On coproducts in varieties, quasivarieties and prevarieties
If the free algebra on one generator in a variety 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 generates the variety .
Generalizing the argument, it is shown that if we are given an algebra and subalgebras, , in a prevariety (-closed class of algebras) such that generates , and also subalgebras such that for each the subalgebra of generated by and is their coproduct in , then the subalgebra of generated by is the coproduct in of these algebras.
Some further results on coproducts are noted:
If satisfies the amalgamation property, then one has the stronger “transitivity” statement, that if has a finite family of subalgebras such that the subalgebra of generated by the is their coproduct, and each has a finite family of subalgebras with the same property, then the subalgebra of generated by all the is their coproduct.
For a residually small prevariety or an arbitrary quasivariety, relationships are proved between the least number of algebras needed to generate as a prevariety or quasivariety, and behavior of the coproduct operation in .
It is shown by example that for a subgroup of the group of all permutations of an infinite set , the group need not have a subgroup isomorphic over to the coproduct with amalgamation . But under various additional hypotheses on , the question remains open.
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
Permanent link to this document
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. https://projecteuclid.org/euclid.ant/1513797499