Vol. 3, No. 8, 2009

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On coproducts in varieties, quasivarieties and prevarieties

George M. Bergman

Vol. 3 (2009), No. 8, 847–879
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 (S-closed class of algebras) P such that An generates P, and also subalgebras Bi Ai1 (0 < i n) 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 S BS. But under various additional hypotheses on B, the question remains open.

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
Mathematical Subject Classification 2000
Primary: 08B25, 08B26, 08C15
Secondary: 03C05, 08A60, 08B20, 20M30
Milestones
Received: 10 June 2008
Revised: 23 November 2009
Accepted: 26 November 2009
Published: 25 December 2009
Authors
George M. Bergman
University of California, Berkeley
Department of Mathematics
Berkeley CA 94720-3840
United States