Subject: categories: Re: Progressive or linear or ... monoids?

From: Peter Freyd <[email-masked]>
Date: Sat, 25 Mar 2006 08:27:04 -0500 (EST)

Vaughan asks:

  Is the quasivariety of monoids generated by the groups and the free
  monoids finitely based?

The free monoids seem to be red herrings. The question is equivalent
to

  Is the quasivariety of monoids generated by the groups finitely
  based?

(since every free monoid is a submonoid of a group.)

If the answer is known I suspect it would be well-known as a theorem
about which monoids can be embedded in groups. I note that a 46-year-
old paper by S.I.Adyan is still cited.  It establishes just the
special case of cancelation monoids given by single defining
relations.

Vaughn goes on to ask

   How different is the abelian case?  More or fewer axioms?

This case is easy. Again, the free commutative monoids are free
herrings. Just add to the equational theory of commutative monoids the
cancelation principle: xy = xz -> y = z. (Every such monoid appears in
the quasivariety since its reflector into the subcat of abelian groups
is an embedding.)