Re: Progressive or linear or ... monoids?

["George Janelidze" <janelg@telkomsa.net>]

Dear Vaughan,

1. Instead of universal Horn formulas I shall talk about quasiidentities,
which is the same thing in your context. Recall: A quasiidentity is a first
order formula of the form (P1&...&Pn)=>Q, where P1,...,Pn,Q are atomic
formulas (i.e. "equations of terms"). A class of universal algebras is said
to be a quasivariety if it can be axiomatized with quasiidentities.
Quasivarieties can be characterized as classes of algebras closed under
subalgebras, products, and filtered colimits (the only reason for filtered
colimits is that we want n above to be finite; products and filtered
colimits can be replaced here with filtered products).

2. Since the free monoid on a set X can be considered as a submonoid of the
free group on the same X, the quasivariety of monoids generated by groups
contain all free monoids. In other words, every quasiidentity that holds for
all groups, will also hold for all free monoids. That is, asking your
question, forget free monoids!

3. The quasivariety of monoids generated by groups obviously consists
exactly of all those monoids that can be embedded into groups. Hence one
should probably see

Malcev, A. Über die Einbettung von assoziativen Systemen in Gruppen.
(Russian) Rec. Math. [Mat. Sbornik] N.S. 6 (48), (1939). 331--336. MR0002152
(2,7d)

Malcev, A. Über die Einbettung von assoziativen Systemen in Gruppen. II.
(Russian) Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940). 251--264. MR0002895
(2,128b)

4. I do not have these papers here in Cape Town. But I seem to remember that
Mal'tsev had an infinite list of quasi-identities which describes the
quasi-variety of monoids that can be embedded into groups. Furthermore it
seems (I am not sure) that he actually introduced the term "quasiidentity"
exactly for this purpose. The first non-trivial one, which he called
Condition Z, was

(as = bt) & (cs = dt) & (au = bv) => (cu = dv);

this condition obviously holds in every group and therefore in every
submonoid of a group, but he constructed a semigroup with cancellation in
which it does not hold. (Here the difference between monoids and semigroups
is irrelevant here). I do not know much did Mal'tsev know about universal
constructions; once Mac Lane told me that Zariski knew them... Knowing them,
one would of course begin by looking at the universal group with a
homomorphism from a given monoid into it, and taking the list of
quasiidentities from the description of the congruence involved in the
construction of that group.

5. Your idea of xy = x => y = 1 and yx = x => y = 1 is too bad (sorry!),
because it is even weaker than the usual cancellation

xy = xz => y = z and yx = zx => y = z.

For, take the quotient of the free monoid on {x,y} by identifying xx = xy
(and = yx = yy if you like); it satisfies your implications but not the
cancellation.

6. In the abelian case (obviously) it is just one quasiidentity, namely the
cancellation. That is, the quasivariety of abelian monoids that can be
embedded into groups is determined by the axiom x + y = x + z => y = z.

George Janelidze

----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: "categories list" <categories@mta.ca>
Sent: Friday, March 24, 2006 10:08 AM
Subject: categories: Progressive or linear or ... monoids?


> 1.  Is the quasivariety of monoids generated by the groups and the free
> monoids finitely based?
>
> That is, is there a finite set of universal Horn formulas entailing the
> common universal Horn theory of groups and free monoids?
>
> In other words, what do groups and free monoids have in common, besides
> being monoids?
>
> Apart from the (equational) axioms for monoids, the only members of that
> theory I can think of are xy=x -> y=1 and yx=x -> y=1.
>
> 2.  How different is the abelian case?  More or fewer axioms?
>
> Vaughan Pratt
>
>
>
>